## 17Calculus Precalculus - Quadratic Formula

##### 17Calculus

The quadratic formula is easy to remember and easy to use. Many students rely on it to solve lots of problems. It is very useful but it can become a crutch if you don't remember where it comes from, why you are using it and how to use it properly.

If you would like a complete lecture on this topic, we recommend this video from one our favorite instructors.

video by Prof Leonard

Here is a playlist of videos that we think will help you.

The quadratic formula is derived by completing the square.   If you know how to complete the square, you do not need the quadratic formula and you can solve many more types of problems than with the quadratic formula alone.

Everywhere you can use the quadratic formula, you can also complete the square.   However, there will be times when the quadratic formula CANNOT be used but completing the square will work.

We highly recommend that you completely understand how to derive the quadratic formula since some instructors may not let you use it.   They may require you to complete the square and show your work.   So make sure you understand how to derive it.

The values of $$x$$ that solve $$ax^2 + bx + c = 0$$ are

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Completing the square is a much more important and versatile technique than using the quadratic formula.

Intuitive Understanding of the Quadratic Formula

Before we get into deriving the quadratic formula, let's watch this video that will help you to intuitively understand where the quadratic formula comes from.

### MindYourDecisions - The Quadratic Formula - Why Do We Complete The Square? INTUITIVE PROOF

Given a quadratic equation $$ax^2 + bx + c = 0$$, we can solve this by completing the square as follows.

 $$ax^2 + bx + c = 0$$ Divide both sides by a. $$x^2 + (b/a)x + (c/a) = 0$$ $$x^2 + (b/a)x = - (c/a)$$ Take the term in front of the x and divide by 2 and square it. Then add that term to both sides of the equation. This is called completing the square. $$x^2 + (b/a)x + [b/(2a)]^2 = - (c/a) + [b/(2a)]^2$$ On the left, we now have a perfect square. $$[x+b/(2a)]^2 = - (c/a) + [b/(2a)]^2$$ Take the square root of both sides, remembering to have plus/minus on one side. [complete the square] $$x+b/(2a) = \pm \sqrt{ - (c/a) + [b/(2a)]^2 }$$ $$x = -b/(2a) \pm \sqrt{ - (c/a) + [b/(2a)]^2 }$$ Let's look at the term under the square root. $$\displaystyle{ \sqrt{ - (c/a) + [b/(2a)]^2 } }$$ $$\displaystyle{ \sqrt{ [b/(2a)]^2 - (c/a) } }$$ $$\displaystyle{ \sqrt{ \left[ \frac{b}{2a} \right]^2 - \frac{c}{a} } }$$ $$\displaystyle{ \sqrt{ \frac{b^2}{4a^2} - \frac{c}{a} } }$$ $$\displaystyle{ \sqrt{ \frac{b^2}{4a^2} - \frac{4ac}{4a^2} } }$$ $$\displaystyle{ \sqrt{ \frac{b^2-4ac}{4a^2} } }$$ $$\displaystyle{ \frac{\sqrt{b^2-4ac}}{\sqrt{4a^2}} }$$ $$\displaystyle{ \frac{\sqrt{b^2-4ac}}{2a} }$$ Now let's put this back into the equation and see what we have. $$\displaystyle{ x = \frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a} }$$ $$\displaystyle{ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} }$$

This last equation is the quadratic formula.
Now, in order to learn this, we recommend that you start from $$ax^2 + bx + c = 0$$ and try to get the quadratic equation on your own without looking at this derivation.   Do this several times until you know it.   It will also help for you to watch a video or two of someone doing this derivation.   The 17Calculus YouTube playlist in the resources panel has several to choose from.   Watch at least one or two of them before working the practice problems.

Practice

Unless otherwise instructed, use the quadratic formula to solve these equations. Check your answers by completing the square.

$$x^2 + x - 12 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$x^2 + x - 12 = 0$$. Check your answer(s) by completing the square.

Solution

### 3078 video solution

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$$x^2 - 6x + 5 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$x^2 - 6x + 5 = 0$$. Check your answer(s) by completing the square.

Solution

### 3079 video solution

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$$(1/4)x^2 - 2x + 3 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$(1/4)x^2 - 2x + 3 = 0$$. Check your answer(s) by completing the square.

Solution

### 3080 video solution

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$$x^2 + 8x - 16 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$x^2 + 8x - 16 = 0$$. Check your answer(s) by completing the square.

Solution

### 3081 video solution

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$$4x^2 - 12x + 9 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$4x^2 - 12x + 9 = 0$$. Check your answer(s) by completing the square.

Solution

### 3082 video solution

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$$3x^2 = 6x-1$$

Problem Statement

Use the quadratic formula to solve the equation $$3x^2 = 6x-1$$. Check your answer(s) by completing the square.

Solution

### Freshmen Math Doctor - 2563 video solution

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$$x^2 + 5x + 2 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$x^2 + 5x + 2 = 0$$. Check your answer(s) by completing the square.

Solution

### Freshmen Math Doctor - 2564 video solution

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$$x^2 - 5x - 6 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$x^2 - 5x - 6 = 0$$. Check your answer(s) by completing the square.

Solution

### PatrickJMT - 2565 video solution

video by PatrickJMT

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$$2x^2 - 4x + 7 = 0$$

Problem Statement

Use the quadratic formula to solve the equation $$2x^2 - 4x + 7 = 0$$. Check your answer(s) by completing the square.

Solution

### PatrickJMT - 2566 video solution

video by PatrickJMT

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