Quadratic Formula Applications

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Q1. Find the roots of the equation x^2 + 2x - 8 = 0.

Solution:

Factoring gives (x + 4)(x - 2) = 0. Thus, the roots are x = 4 and x = -2.

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Q2. What are the roots of the polynomial equation x^2 - 5x + 6 = 0?

Solution:

Factoring gives (x - 2)(x - 3) = 0. Thus, the roots are x = 2 and x = 3.

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Q3. What is the solution to the equation 5x^2 - 20 = 0?

Solution:

First, add 20 to both sides: 5x^2 = 20. Then divide by 5: x^2 = 4. Taking the square root gives x = 4 and x = -4.

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Q4. Solve the quadratic equation 3x^2 + 12x + 12 = 0.

Solution:

Dividing the entire equation by 3 gives x^2 + 4x + 4 = 0, which factors to (x + 2)(x + 2) = 0. Thus, x = -2.

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Q5. What is the solution to the equation x^2 - 7x + 10 = 0?

Solution:

Factoring gives (x - 5)(x - 2) = 0. Thus, x = 5 and x = 2.

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Q6. Find the roots of the quadratic equation 3x^2 - 12x = 0.

Solution:

Factoring gives 3x(x - 4) = 0. Thus, x = 0 and x = 4.

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Q7. If 2x + 3 > 7, what is the solution set for x?

Solution:

Subtract 3 from both sides: 2x > 4. Divide by 2: x > 2.

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Q8. Solve for x in the equation x^2 - 4 = 0.

Solution:

Factoring gives (x - 2)(x + 2) = 0. Thus, the solutions are x = 2 and x = -2.

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Q9. Solve for x in the equation 5x^2 + 10x = 0.

Solution:

Factoring gives 5x(x + 2) = 0. Thus, x = 0 or x + 2 = 0, giving x = -2.

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Q10. What is the solution to the equation 4x^2 + 8x + 4 = 0?

Solution:

This can be factored as 4(x^2 + 2x + 1) = 0, which simplifies to 4(x + 1)(x + 1) = 0. Thus, x = -1.