Math: Quadratic Formula Tune Practice Worksheet

Question 1

Solve x^2 + 5x + 6 = 0 using the quadratic formula.

Accepted Answer

Here a = 1, b = 5, and c = 6. The quadratic formula gives x = (-5 ± sqrt(25 - 24)) / 2 = (-5 ± 1) / 2, so the solutions are x = -2 and x = -3.

Question 2

Solve 2x^2 - 3x - 2 = 0 using the quadratic formula.

Accepted Answer

Here a = 2, b = -3, and c = -2. The formula gives x = (3 ± sqrt(9 + 16)) / 4 = (3 ± 5) / 4, so the solutions are x = 2 and x = -1/2.

Question 3

Solve x^2 - 4x + 7 = 0 using the quadratic formula. State whether the solutions are real or complex.

Accepted Answer

Here a = 1, b = -4, and c = 7. The formula gives x = (4 ± sqrt(16 - 28)) / 2 = (4 ± sqrt(-12)) / 2 = 2 ± i sqrt(3), so the solutions are complex.

Question 4

For the equation 3x^2 + 6x + 3 = 0, use the discriminant to predict the number of real solutions. Then solve.

Accepted Answer

Here a = 3, b = 6, and c = 3. The discriminant is 6^2 - 4(3)(3) = 36 - 36 = 0, so there is one real solution. The solution is x = -6 / 6 = -1.

Question 5

Solve 4x^2 + 4x - 3 = 0 using the quadratic formula.

Accepted Answer

Here a = 4, b = 4, and c = -3. The formula gives x = (-4 ± sqrt(16 + 48)) / 8 = (-4 ± 8) / 8, so the solutions are x = 1/2 and x = -3/2.

Question 6

Solve 5x^2 - 2x + 1 = 0 using the quadratic formula. Write the answer in simplest complex form.

Accepted Answer

Here a = 5, b = -2, and c = 1. The formula gives x = (2 ± sqrt(4 - 20)) / 10 = (2 ± sqrt(-16)) / 10 = (2 ± 4i) / 10, so x = (1 ± 2i) / 5.

Question 7

A student is solving 2x^2 + 7x + 3 = 0 and writes x = (-7 ± sqrt(7^2 - 4(2)(3))) / 2. Explain the mistake and write the correct setup.

Accepted Answer

The mistake is that the denominator should be 2a, not just 2. Since a = 2, the correct setup is x = (-7 ± sqrt(7^2 - 4(2)(3))) / 4.

Question 8

Solve 2x^2 + 7x + 3 = 0 using the correct quadratic formula setup.

Accepted Answer

Here a = 2, b = 7, and c = 3. The formula gives x = (-7 ± sqrt(49 - 24)) / 4 = (-7 ± 5) / 4, so the solutions are x = -1/2 and x = -3.

Question 9

The graph of a quadratic equation crosses the x-axis at x = 1 and x = 4. What are the solutions of the equation, and what does this mean about the discriminant?

Accepted Answer

The solutions are x = 1 and x = 4 because the x-intercepts are the zeros of the quadratic equation. Since there are two real solutions, the discriminant is positive.

Question 10

Solve -x^2 + 6x - 8 = 0 using the quadratic formula.

Accepted Answer

Here a = -1, b = 6, and c = -8. The formula gives x = (-6 ± sqrt(36 - 32)) / -2 = (-6 ± 2) / -2, so the solutions are x = 2 and x = 4.

Question 11

Find the exact solutions of x^2 + 2x - 5 = 0 using the quadratic formula.

Accepted Answer

Here a = 1, b = 2, and c = -5. The formula gives x = (-2 ± sqrt(4 + 20)) / 2 = (-2 ± sqrt(24)) / 2 = (-2 ± 2sqrt(6)) / 2, so x = -1 ± sqrt(6).

Question 12

A ball's height in meters is modeled by h = -5t^2 + 20t + 1. Use the quadratic formula to find when the ball hits the ground. Round your answer to the nearest hundredth.

Accepted Answer

Set h = 0 to get -5t^2 + 20t + 1 = 0. Using a = -5, b = 20, and c = 1 gives t = (-20 ± sqrt(400 + 20)) / -10 = (-20 ± sqrt(420)) / -10. The positive time is about 4.05 seconds, so the ball hits the ground after about 4.05 seconds.

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Math: Quadratic Formula Tune Practice

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