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## Solve quadratic equations using the formula

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Solve the following quadratic equation algebraically:

### Guidance

You can use the method of completing the square to solve the general quadratic equation . The result will be a formula that you can use to solve any quadratic equation given the values for and . The following is a derivation of the quadratic formula:

Step 1: Divide the general equation by . Then, move the third term on the left side to the right side of the equation.

Step 2: Complete the square. Note that your "" value in this case is actually .

Step 3: Simplify.

Step 4: Rewrite the left side of the equation as a binominal squared. Then, take the square root of both sides and solve for .

This is known as the quadratic formula. You can use the quadratic formula to solve ANY quadratic equation. All you need to know are the values of and . Keep in mind that while the factoring method for solving a quadratic equation will only sometimes work, the quadratic formula will ALWAYS work. You should memorize the quadratic formula because you will use it in algebra and future math courses.

#### Example A

Find the exact solutions of the following quadratic equation using the quadratic formula:

Solution: For this quadratic equation, . Substitute these values into the quadratic formula and simplify.

The exact solutions to the quadratic equation are .

#### Example B

Use the quadratic formula to determine the approximate solutions of the equation:

Solution: Start by rewriting the equation in standard form so that it is set equal to zero. becomes . For this quadratic equation, . Substitute these values into the quadratic formula and simplify.

The approximate solutions to the quadratic equation to the nearest tenth are or .

#### Example C

Solve the following equation using the quadratic formula:

Solution: While this does not look like a quadratic equation (it is actually a rational equation because it contains rational expressions), you can rewrite it as a quadratic equation by multiplying by to get rid of the fractions. Note that and are the denominators you want to eliminate. This is why you want to multiply by . After multiplying, simplify and put the equation in standard quadratic form set equal to 0.

For this quadratic equation, . Substitute these values into the quadratic formula and simplify.

The exact solutions to the equation are or . Note that neither of these solutions will cause the original equation to have a zero in the denominator, so they both work.

#### Concept Problem Revisited

To solve the equation algebraically, you can use the quadratic formula. For this quadratic equation, . Substitute these values into the quadratic formula and simplify.

The solutions are .

### Vocabulary

The quadratic formula is the formula used to determine the solutions for a quadratic equation.

### Guided Practice

1. For the following equation, rewrite as a quadratic equation and state the values for and :

3. Find the approximate solutions to the following equation:

1. Multiply by to clear the fractions.

For this equation, .

2. This equation does not have a ‘’ term. The value of ‘’ is 0. For this equation, .

3. Multiply by to clear the fractions.

For this equation, .

The solutions to the quadratic equation to the nearest tenth are or .

### Practice

State the value of and for each of the following quadratic equations.

Determine the exact roots of the following quadratic equations using the quadratic formula.

### Vocabulary Language: English

Binomial

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.
Completing the Square

Completing the Square

Completing the square is a common method for rewriting quadratics. It refers to making a perfect square trinomial by adding the square of 1/2 of the coefficient of the $x$ term.

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
Roots

Roots

The roots of a function are the values of x that make y equal to zero.
Square Root

Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.
Vertex

Vertex

The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward.