**Introduction to Quadratic Equations**

A quadratic equation is one that can be put in the form *ax* ^{2} + *bx* + *c* = 0 where *a* , *b* , and *c* are numbers and *a* is not zero ( *b* and/or *c* might be zero). For instance 3 *x* ^{2} + 7 *x* = 4 is a quadratic equation.

3 *x* ^{2} + 7 *x* =4

– 4 – 4

3 *x* ^{2} + 7 *x* – 4 = 0

In this example *a* = 3, *b* = 7, and *c* =–4.

**Solving Quadratic Equations - When the Product of Two Numbers is Zero**

There are two main approaches to solving these equations. One approach uses the fact that if the product of two numbers is zero, at least one of the numbers must be zero. In other words, *wz* = 0 implies *w* = 0 or *z* = 0 (or both *w* = 0 and *z* = 0.) To use this fact on a quadratic equation first make sure that one side of the equation is zero and factor the other side. Set each factor equal to zero then solve for *x* .

**Examples**

**Example 1:**

*x* ^{2} + 2 *x* – 3 = 0

*x* ^{2} + 2 *x* – 3 can be factored as ( *x* + 3)( *x* – 1)

*x* ^{2} + 2 *x* – 3 = 0 becomes ( *x* + 3)( *x* –1)= 0

Now set each factor equal to zero and solve for *x* .

You can check your solutions by substituting them into the original equation.

**Example 2:**

**Example 3:**

**Example 4:**

3 *x* ^{2} − 9 *x* −30 = 0 becomes 3( *x* ^{2} − 3 *x* −10) = 0 which becomes 3( *x* − 5) ( *x* + 2) = 0

The factor 3 was not set equal to zero because “3 = 0” does not lead to any solution.

Find practice problems and solutions at Quadratic Equations Practice Problems - Set 1.

**Solving Quadratic Equations - Multiplying and Dividing Both Sides of the Equation**

Not all quadratic expressions will be as easy to factor as the previous examples and problems were. Sometimes you will need to multiply or divide both sides of the equation by a number. Because zero multiplied or divided by any nonzero number is still zero, only one side of the equation will change. Keep in mind that not all quadratic expressions can be factored using rational numbers (fractions) or even real numbers. Fortunately there is another way of solving quadratic equations, which bypasses the factoring method.

**Example**

The equation – *x* ^{2} + 4 *x* – 3 = 0 is awkward to factor because of the negative sign in front of x ^{2}. Multiply both sides of the equation by – 1 then factor.

–1(–x^{2} + 4 *x* – 3) = –1(0)

x^{2} – 4 *x* + 3 = 0

( *x* –3)( *x* –1) = 0

**Multiplying Both Sides of the Equation - Eliminating Decimals and Fractions**

Decimals and fractions in a quadratic equation can be eliminated in the same way. Multiply both sides of the equation by a power of 10 to eliminate decimal points. Multiply both sides of the equation by the LCD to eliminate fractions.

**Examples**

**Example 1:**

0.1 *x* ^{2} – 1.5 *x* + 5.6 = 0

Multiply both sides of the equation by 10 to clear the decimal.

**Example 2:**

Clear the fraction by multiplying both sides of the equation by 4 (the LCD).

**Example 3:**

**Example 4:**

Multiplying both sides of the equation by would have combined two steps.

Find practice problems and solutions at Quadratic Equations Practice Problems - Set 2.

**Solving Quadratic Equations - Simplifying Squares and Roots**

Sometimes using the fact that *x* ^{2} = *k* implies can be used to solve quadratic equations. For instance, if x ^{2} = 9, then *x* = 3 or – 3 because 3 ^{2} = 9 and (– 3) ^{2} = 9. This method works if the equation can be put in the form *ax* ^{2} – *c* = 0, where *a* and *c* are not negative.

**Examples**

**Example 1:**

**Example 2:**

25 – *x* ^{2} = 0

25 = *x* ^{2}

±5 = *x*

**Example 3:**

**Example 4:**

Find practice problems and solutions at Quadratic Equations Practice Problems - Set 3.

More practice problems for this concept can be found at: Algebra Quadratic Equations Practice Test.

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