Lesson 37, Quadratic equations: Section 2Proof of the quadratic formula IN LESSON 18 we saw a technique called completing the square. We will now apply it to solving a quadratic equation. Completing the square If we try to solve this quadratic equation by factoring,
This method is valid only when the coefficient of x^{2} is 1. 1) Transpose the constant term to the right x^{2} + 6x = −2.
x^{2} + 6x + 9 = −2 + 9. The left-hand side is now the perfect square of (x + 3). (x + 3)^{2} = 7. That has the form
The solutions are the conjugate pair: −3 + , −3 − . For a method of checking those roots, see the theorem of the sum and product of the roots: Lesson 10 of Topics in Precalculus, In Lesson 18 there are examples and problems in which the coefficient of x is odd. Also, some of the quadratics below have complex roots, and some involve simplifying radicals. Problem 6. Solve each quadratic equation by completing the square. To see the answer, pass your cursor from left to right
Problem 7. Find two numbers whose sum is 10 and whose product is 20. x = 5 ± According to the theorem of the sum and products of the roots, they are the solutions to Problem 6b above. The quadratic formula Here is a formula for finding the roots of any quadratic. It is proved by completing the square In other words, the quadratic formula completes the square for us. Theorem. If ax^{2} + bx + c = 0, then The two roots are on the right. One root has the plus sign; the other, the minus sign. If the square root term is irrational, then the two roots are a conjugate pair. If we call those two roots r_{1} and r_{2} , then the quadratic can be factored as (x − r_{1})(x − r_{2}). We will prove the quadratic formula below. Example 4. Use the quadratic formula to solve this quadratic equation: 3x^{2} + 5x − 8 = 0 Solution. We have: a = 3, b = 5, c = −8. Therefore, according to the formula:
That is,
Those are the two roots. And they are rational. When the roots are rational, we could have solved the equation by factoring, which is always the simplest method.
Problem 8. Use the quadratic formula to find the roots of each quadratic. a) x^{2} − 5x + 5 a = 1, b = −5, c = 5.
b) 2x^{2} − 8x + 5 a = 2, b = −8, c = 5.
c) 5x^{2} − 2x + 2 a = 5, b = −2, c = 2.
The discriminant The radicand b^{2} − 4ac is called the discriminant. If the discriminant is
Problem 9. Show: If the roots of ax^{2} + bx + c are complex, and a, b, c are positive, then 2a − b + c > 0. Since the roots are complex, then the discriminant b^{2} − 4ac < 0. That implies b^{2} < 4ac. Now, 2a − b + c > 0 if and only if b < 2a + c if and only if b^{2} < 4a^{2} + 4ac + c^{2}, which is true. For, since b^{2} < 4ac, it is less than 4ac plus a positive quantity. 4a^{2} and c^{2} are positive. The student should be familiar with the logical expression Proof of the quadratic formula To prove the quadratic formula, we complete the square. But to do that, the coefficient of x^{2} must be 1. Therefore, we will divide both sides of the original equation by a: on multiplying both c and a by 4a, thus making the denominators the same (Lesson 23), This is the quadratic formula. Section 3: The graph of y = A quadratic Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |