9.1 Solving Quadratic Equations by Roots and Factoring

9.1 Solving Quadratic Equations by Roots and Factoring

Quick Answer: A quadratic equation in one variable is any equation that can be written in standard form ax^2 + bx + c = 0 with a != 0. On the Florida Algebra 1 EOC (B.E.S.T.), MA.912.AR.3.1 expects you to solve quadratics by isolating x^2 and taking square roots, and MA.912.AR.3.4 expects you to solve by factoring and applying the zero-product property. Picking the faster method for the equation in front of you is a test-day skill the EOC rewards.

Standard Form and What "Solving" Means

A quadratic equation in one variable has standard form ax^2 + bx + c = 0, where a, b, c are real and a != 0. Solving means finding every value of x (the roots) that makes the equation true. On the Algebra 1 EOC you report real roots only; some methods reveal two distinct roots, some reveal one repeated root, and some reveal no real roots.

Method selection:

• x^2 = k or ax^2 = k (b = 0): square-root property.
• ax^2 + bx + c = 0, factorable over integers: factoring.
• ax^2 + bx + c = 0, not factorable: quadratic formula or completing the square (Section 9.2).

Method 1: Square-Root Property (MA.912.AR.3.1)

When the quadratic has no linear term (b = 0), isolate x^2 and take the square root of both sides. A square root has two values, positive and negative, so write +/- explicitly.

Worked example. Solve x^2 - 50 = 0. Add 50: x^2 = 50. Take the square root: x = +/- sqrt(50). Simplify: sqrt(50) = 5 sqrt(2). Solution set: {-5 sqrt(2), 5 sqrt(2)}.

Worked example with a coefficient. Solve 3x^2 - 27 = 0. Add 27: 3x^2 = 27. Divide by 3: x^2 = 9. Take square roots: x = +/- 3.

EOC traps with the square-root method

• Forgetting +/-. Writing x = sqrt(9) = 3 loses the negative root. Include +/- whenever you take the square root of a variable expression that is not known to be nonnegative.
• Taking the root before isolating x^2. For 3x^2 = 27 you cannot write x = sqrt(27); divide by 3 first.
• Leaving the radical unsimplified. The B.E.S.T. standards require simplified radical form (MA.912.NSO.1.4), so sqrt(50) must be 5 sqrt(2).
• No real roots. If isolation gives x^2 = -16, there is no real solution. Report "no real roots"; complex roots are outside the Algebra 1 course.

Method 2: Factoring and the Zero-Product Property (MA.912.AR.3.4)

The zero-product property says: if a * b = 0, then a = 0 or b = 0. To use it, the quadratic must equal zero first. If you are given x^2 - 5x = 6, rewrite it as x^2 - 5x - 6 = 0 before factoring; many students factor the left side of an equation that still has a constant on the right and lose both roots.

Worked example. Solve x^2 - 5x - 6 = 0. Find two numbers whose product is -6 and sum is -5: -6 and +1. Write (x - 6)(x + 1) = 0. Zero-product gives x = 6 or x = -1.

Factoring patterns to recognize on sight

• GCF first: 2x^2 + 8x = 2x(x + 4).
• x^2 + bx + c (a = 1): two numbers sum to b, product to c. Example: x^2 + 7x + 12 = (x + 3)(x + 4).
• ax^2 + bx + c (a != 1): AC method. For 3x^2 + 11x + 6, a*c = 18; the pair 9 and 2 splits the middle: 3x^2 + 9x + 2x + 6 = 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3).
• Difference of squares a^2 - b^2: (a - b)(a + b). Example: x^2 - 49 = (x - 7)(x + 7).
• Perfect square trinomial a^2 +/- 2ab + b^2: (a +/- b)^2. Example: x^2 + 10x + 25 = (x + 5)^2.

When factoring is not enough

Factoring over the integers works cleanly only when the roots are rational. For x^2 - 5 = 0, the left side factors as (x - sqrt(5))(x + sqrt(5)) but the square-root method is faster. For x^2 + x - 1 = 0, the expression does not factor over the integers; you must use the quadratic formula (Section 9.2). Before factoring, glance at the discriminant b^2 - 4ac: if it is a perfect square (and a = 1, or a divides evenly), the quadratic is factorable over the rationals.

Real-World Context Trap

EOC word problems often ask you to set up the quadratic before solving. A rectangle has length x and width (12 - x); area = x(12 - x) = 12x - x^2. If area must equal 20, write 12x - x^2 = 20, then move everything to one side: x^2 - 12x + 20 = 0. Only now can you factor: (x - 10)(x - 2) = 0, so x = 2 or x = 10. Both are valid because both are positive and less than 12. Always check each root against the context's constraints before reporting.

Method-Selection Checklist

Before you solve: (1) Is the equation equal to zero? (2) Is b = 0? Use the square-root property. (3) Does it factor over the integers? Factor and use zero-product. (4) Otherwise, go to completing the square or the quadratic formula (Section 9.2). This prevents the two most common EOC errors: applying the zero-product property to an equation not equal to zero, and factoring a quadratic whose discriminant is negative.