Solving Quadratic Equations
This lesson introduces solving quadratic equations by the square root method, completing the square, factoring, and using the quadratic formula.
Solving Quadratic Equations by the Square Root Method
A quadratic equation is an equation where the highest variable is squared. The equation is in the form \(ax^2 + bx + c = 0\), where a is a non-zero constant and b and c are constants. There are at most two solutions to the equation because the highest variable is squared. There are many methods to solve a quadratic equation.
This section will explore solving a quadratic equation by the square root method. The equation must be in the form of \(ax^2 = c\), or there is no x term.
STEP BY STEP
Step 1. Use multiplication and division properties of an equation to remove the value in front of the variable.
Step 2. Apply the square root to both sides of the equation.
Note: The positive and negative square root make the solution true. For the equation \(x^2 = 9\), the solutions are \(–3\) and \(3\) because \(3^2 = 9\) and \((–3)^2 = 9\).
Example
Solving Quadratic Equations Using the Square Root Method Review
Solving Quadratic Equations by Completing the Square
A quadratic equation in the form \(x^2 + bx\) can be solved by a process known as completing the square. The best time to solve by completing the square is when the b term is even.
STEP BY STEP
Step 1. Divide all terms by the coefficient of \(x ^2\).
Step 2. Move the number term to the right side of the equation.
Step 3. Complete the square \((\frac{b }{2 }) ^2\) and add this value to both sides of the equation.
Step 4. Factor the left side of the equation.
Step 5. Apply the square root to both sides of the equation.
Step 6. Use addition and subtraction properties to move all number terms to the right side of the equation.
Examples
Solving Quadratic Equations By Completing the Square Review
Solving Quadratic Equations by Factoring
Factoring can only be used when a quadratic equation is factorable; other methods are needed to solve quadratic equations that are not factorable.
STEP BY STEP
Step 1. Simplify if needed by clearing any fractions and parentheses.
Step 2. Write the equation in standard form, \(ax^ 2 + bx + c = 0\).
Step 3. Factor the quadratic equation.
Step 4. Set each factor equal to zero.
Step 5. Solve the linear equations using inverse operations.
The quadratic equation will have two solutions if the factors are different or one solution if the factors are the same.
Examples
Solving Quadratic Equations by Factoring Review
Solving Quadratic Equations by the Quadratic Formula
Many quadratic equations are not factorable. Another method of solving a quadratic equation is by using the quadratic formula. This method can be used to solve any quadratic equation in the form . Using the coefficients a, b, and c, the quadratic formula is \(x = \frac{–b±\sqrt{b^2–4ac}}{2a}\). The values are substituted into the formula, and applying the order of operations finds the solution(s) to the equation.
KEEP IN MIND
Watch the negative sign in the formula. Remember that a number squared is always positive.
The solution of the quadratic formula in these examples will be exact or estimated to three decimal places. There may be cases where the exact solutions to the quadratic formula are used.
Examples
Solving Quadratic Equations Using the Quadratic Formula Review
Let’s Review!
- There are four methods to solve a quadratic equation algebraically:
- The square root method is used when there is a squared variable term and a constant term.
- Completing the square is used when there is a squared variable term and an even variable term.
- Factoring is used when the equation can be factored.
- The quadratic formula can be used for any quadratic equation.
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