## Lesson 37, Quadratic equations: Section 2Proof of the quadratic formula IN LESSON 18 we saw a technique called completing the square. We will now see how to apply it to solving a quadratic equation. Completing the square If we try to solve this quadratic equation by factoring,
The technique is valid only when the coefficient of 1) Transpose the constant term to the right
The left-hand side is now the perfect square of ( ( 3 is That equation has the form
That is, the solutions to
are the conjugate pair, −3 + , −3 − . For a method of checking these 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 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. According to the theorem of the sum and products of the roots, they are the solutions to Problem 6b. 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.
Theorem. 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 ( We will prove the quadratic formula below. Example 4. Use the quadratic formula to solve this quadratic equation: 3
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)
b) 2
c) 5
The discriminant The radicand
Proof of the quadratic formula To prove the quadratic formula, we complete the square. But to do that, the coefficient of on multiplying both This is the quadratic formula. Section 3: The graph of Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |