Solve Quadratic Equations by Factoring or the Quadratic Formula

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What is a quadratic equation?

Back

A quadratic equation is a polynomial equation of degree 2, typically in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

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What does it mean to solve a quadratic equation by factoring?

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Solving a quadratic equation by factoring involves expressing the equation in the form (px + q)(rx + s) = 0 and finding the values of x that make each factor equal to zero.

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What is the Quadratic Formula?

Back

The Quadratic Formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions of a quadratic equation ax² + bx + c = 0.

Front

What is the discriminant in a quadratic equation?

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The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots: if positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.

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How do you find the roots of the equation x² - 6x + 5 = 0?

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Factor the equation as (x - 5)(x - 1) = 0, giving roots x = 5 and x = 1.

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What are the steps to solve a quadratic equation using the Quadratic Formula?

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1. Identify coefficients a, b, and c. 2. Calculate the discriminant (b² - 4ac). 3. Substitute a, b, and the discriminant into the Quadratic Formula. 4. Simplify to find the values of x.

Front

What is the vertex form of a quadratic equation?

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The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

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How can you determine if a quadratic equation can be factored?

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A quadratic equation can be factored if its discriminant (b² - 4ac) is a perfect square, indicating rational roots.

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What is the significance of the coefficient 'a' in a quadratic equation?

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The coefficient 'a' determines the direction of the parabola: if a > 0, it opens upwards; if a < 0, it opens downwards.

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What does it mean for a quadratic equation to have no real solutions?

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A quadratic equation has no real solutions when its discriminant is negative (b² - 4ac < 0), indicating that the roots are complex.