Quadratic Equation worksheet

1. Roots: x = 3.5 or x = −1

1. Start with your equation set equal to zero, where a = 2, b = −5 and c = −7

   $\text{2x² − 5x − 7 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-5}\right)}^2-4\times 2\times \left(\mathrm{-7}\right)$$=25-\left(\mathrm{-56}\right)$$=81$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{5\pm \sqrt{81}}{2\times 2}$$=\frac{5\pm 9}{4}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{5+9}{4}$$=3.5$

6. Then use the − sign for the other root

   $\frac{5-9}{4}$$=\mathrm{-1}$

7. So the two roots are

   $\text{x = 3.5 or x = −1}$

8. Written exactly, without rounding, the roots are fractions

   $\text{x = 7/2 or x = −1}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{5}{2\times 2}$$=1.25$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-7}-\frac{{\left(\mathrm{-5}\right)}^2}{4\times 2}$$=\mathrm{-10.125}$

11. Because the roots are whole numbers or fractions, the left side can be split into factors. Each root r gives a factor (x − r), scaled so the numbers match

    $\text{(2x − 7)(x + 1) = 0}$

2. Roots: x = 2.350781 or x = −0.850781

1. Start with your equation set equal to zero, where a = 2, b = −3 and c = −4

   $\text{2x² − 3x − 4 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-3}\right)}^2-4\times 2\times \left(\mathrm{-4}\right)$$=9-\left(\mathrm{-32}\right)$$=41$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{3\pm \sqrt{41}}{2\times 2}$$=\frac{3\pm 6.403124}{4}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{3+6.403124}{4}$$=2.350781$

6. Then use the − sign for the other root

   $\frac{3-6.403124}{4}$$=\mathrm{-0.850781}$

7. So the two roots are

   $\text{x = 2.350781 or x = −0.850781}$

8. To write the roots exactly, without rounding: simplify √41 = √41 and divide out any shared factor

   $\text{x = (3 ± √41)/4}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{3}{2\times 2}$$=0.75$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-4}-\frac{{\left(\mathrm{-3}\right)}^2}{4\times 2}$$=\mathrm{-5.125}$

3. Roots: x = 2.5 or x = −1

1. Start with your equation set equal to zero, where a = 2, b = −3 and c = −5

   $\text{2x² − 3x − 5 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-3}\right)}^2-4\times 2\times \left(\mathrm{-5}\right)$$=9-\left(\mathrm{-40}\right)$$=49$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{3\pm \sqrt{49}}{2\times 2}$$=\frac{3\pm 7}{4}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{3+7}{4}$$=2.5$

6. Then use the − sign for the other root

   $\frac{3-7}{4}$$=\mathrm{-1}$

7. So the two roots are

   $\text{x = 2.5 or x = −1}$

8. Written exactly, without rounding, the roots are fractions

   $\text{x = 5/2 or x = −1}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{3}{2\times 2}$$=0.75$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-5}-\frac{{\left(\mathrm{-3}\right)}^2}{4\times 2}$$=\mathrm{-6.125}$

11. Because the roots are whole numbers or fractions, the left side can be split into factors. Each root r gives a factor (x − r), scaled so the numbers match

    $\text{(2x − 5)(x + 1) = 0}$

4. Roots: x = 3.345208 or x = −1.345208

1. Start with your equation set equal to zero, where a = 2, b = −4 and c = −9

   $\text{2x² − 4x − 9 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-4}\right)}^2-4\times 2\times \left(\mathrm{-9}\right)$$=16-\left(\mathrm{-72}\right)$$=88$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{4\pm \sqrt{88}}{2\times 2}$$=\frac{4\pm 9.380832}{4}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{4+9.380832}{4}$$=3.345208$

6. Then use the − sign for the other root

   $\frac{4-9.380832}{4}$$=\mathrm{-1.345208}$

7. So the two roots are

   $\text{x = 3.345208 or x = −1.345208}$

8. To write the roots exactly, without rounding: simplify √88 = 2√22 and divide out any shared factor

   $\text{x = (2 ± √22)/2}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{4}{2\times 2}$$=1$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-9}-\frac{{\left(\mathrm{-4}\right)}^2}{4\times 2}$$=\mathrm{-11}$

5. Roots: x = 2.573384 or x = −0.906718

1. Start with your equation set equal to zero, where a = 3, b = −5 and c = −7

   $\text{3x² − 5x − 7 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-5}\right)}^2-4\times 3\times \left(\mathrm{-7}\right)$$=25-\left(\mathrm{-84}\right)$$=109$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{5\pm \sqrt{109}}{2\times 3}$$=\frac{5\pm 10.440307}{6}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{5+10.440307}{6}$$=2.573384$

6. Then use the − sign for the other root

   $\frac{5-10.440307}{6}$$=\mathrm{-0.906718}$

7. So the two roots are

   $\text{x = 2.573384 or x = −0.906718}$

8. To write the roots exactly, without rounding: simplify √109 = √109 and divide out any shared factor

   $\text{x = (5 ± √109)/6}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{5}{2\times 3}$$=0.833333$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-7}-\frac{{\left(\mathrm{-5}\right)}^2}{4\times 3}$$=\mathrm{-9.083333}$

6. Roots: x = 2 or x = −1

1. Start with your equation set equal to zero, where a = 3, b = −3 and c = −6

   $\text{3x² − 3x − 6 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-3}\right)}^2-4\times 3\times \left(\mathrm{-6}\right)$$=9-\left(\mathrm{-72}\right)$$=81$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{3\pm \sqrt{81}}{2\times 3}$$=\frac{3\pm 9}{6}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{3+9}{6}$$=2$

6. Then use the − sign for the other root

   $\frac{3-9}{6}$$=\mathrm{-1}$

7. So the two roots are

   $\text{x = 2 or x = −1}$

8. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{3}{2\times 3}$$=0.5$

9. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

   $k$$=\mathrm{-6}-\frac{{\left(\mathrm{-3}\right)}^2}{4\times 3}$$=\mathrm{-6.75}$

10. Because the roots are whole numbers or fractions, the left side can be split into factors. Each root r gives a factor (x − r), scaled so the numbers match

    $\text{3(x − 2)(x + 1) = 0}$

7. Roots: x = 2.230139 or x = −0.896805

1. Start with your equation set equal to zero, where a = 3, b = −4 and c = −6

   $\text{3x² − 4x − 6 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-4}\right)}^2-4\times 3\times \left(\mathrm{-6}\right)$$=16-\left(\mathrm{-72}\right)$$=88$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{4\pm \sqrt{88}}{2\times 3}$$=\frac{4\pm 9.380832}{6}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{4+9.380832}{6}$$=2.230139$

6. Then use the − sign for the other root

   $\frac{4-9.380832}{6}$$=\mathrm{-0.896805}$

7. So the two roots are

   $\text{x = 2.230139 or x = −0.896805}$

8. To write the roots exactly, without rounding: simplify √88 = 2√22 and divide out any shared factor

   $\text{x = (2 ± √22)/3}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{4}{2\times 3}$$=0.666667$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-6}-\frac{{\left(\mathrm{-4}\right)}^2}{4\times 3}$$=\mathrm{-7.333333}$

8. Roots: x = 5.162278 or x = −1.162278

1. Start with your equation set equal to zero, where a = 1, b = −4 and c = −6

   $\text{x² − 4x − 6 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-4}\right)}^2-4\times 1\times \left(\mathrm{-6}\right)$$=16-\left(\mathrm{-24}\right)$$=40$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{4\pm \sqrt{40}}{2\times 1}$$=\frac{4\pm 6.324555}{2}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{4+6.324555}{2}$$=5.162278$

6. Then use the − sign for the other root

   $\frac{4-6.324555}{2}$$=\mathrm{-1.162278}$

7. So the two roots are

   $\text{x = 5.162278 or x = −1.162278}$

8. To write the roots exactly, without rounding: simplify √40 = 2√10 and divide out any shared factor

   $\text{x = 2 ± √10}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{4}{2\times 1}$$=2$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-6}-\frac{{\left(\mathrm{-4}\right)}^2}{4\times 1}$$=\mathrm{-10}$

9. Roots: x = 2.870829 or x = −0.870829

1. Start with your equation set equal to zero, where a = 2, b = −4 and c = −5

   $\text{2x² − 4x − 5 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-4}\right)}^2-4\times 2\times \left(\mathrm{-5}\right)$$=16-\left(\mathrm{-40}\right)$$=56$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{4\pm \sqrt{56}}{2\times 2}$$=\frac{4\pm 7.483315}{4}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{4+7.483315}{4}$$=2.870829$

6. Then use the − sign for the other root

   $\frac{4-7.483315}{4}$$=\mathrm{-0.870829}$

7. So the two roots are

   $\text{x = 2.870829 or x = −0.870829}$

8. To write the roots exactly, without rounding: simplify √56 = 2√14 and divide out any shared factor

   $\text{x = (2 ± √14)/2}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{4}{2\times 2}$$=1$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-5}-\frac{{\left(\mathrm{-4}\right)}^2}{4\times 2}$$=\mathrm{-7}$

10. Roots: x = 2.765564 or x = −1.265564

1. Start with your equation set equal to zero, where a = 2, b = −3 and c = −7

   $\text{2x² − 3x − 7 = 0}$

2. The quadratic formula finds the roots: the values of x that make the equation true

   $x$$=\frac{-b\pm \sqrt{D}}{2a}$

3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

   $D$$={\left(\mathrm{-3}\right)}^2-4\times 2\times \left(\mathrm{-7}\right)$$=9-\left(\mathrm{-56}\right)$$=65$

4. D is positive, so there are two real roots. Put your numbers into the formula

   $x$$=\frac{3\pm \sqrt{65}}{2\times 2}$$=\frac{3\pm 8.062258}{4}$

5. The ± means "plus or minus". Use the + sign for one root

   $\frac{3+8.062258}{4}$$=2.765564$

6. Then use the − sign for the other root

   $\frac{3-8.062258}{4}$$=\mathrm{-1.265564}$

7. So the two roots are

   $\text{x = 2.765564 or x = −1.265564}$

8. To write the roots exactly, without rounding: simplify √65 = √65 and divide out any shared factor

   $\text{x = (3 ± √65)/4}$

9. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

   $h$$=\frac{3}{2\times 2}$$=0.75$

10. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    $k$$=\mathrm{-7}-\frac{{\left(\mathrm{-3}\right)}^2}{4\times 2}$$=\mathrm{-8.125}$