Polynomial Equations: Background You’ll Need 4

  • Solve quadratic equations

An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as [latex]2{x}^{2}+3x - 1=0[/latex] and [latex]{x}^{2}-4=0[/latex] are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.

Often the easiest method of solving a quadratic equation is factoring. Factoring a quadratic equation involves expressing it as a product of simpler polynomials, typically two binomials.

The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer.

How To: Given a quadratic equation with the leading coefficient of 1, factor it

  1. Find two numbers whose product equals c and whose sum equals [latex]b[/latex].
  2. Use those numbers to write two factors of the form [latex]\left(x+k\right)\text{ or }\left(x-k\right)[/latex], where [latex]k[/latex] is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are [latex]1[/latex] and [latex]-2[/latex], the factors are [latex]\left(x+1\right)\left(x - 2\right)[/latex].
  3. Solve using the zero-product property by setting each factor equal to zero and solving for the variable.
Factor and solve the equation: [latex]{x}^{2}+x - 6=0[/latex].

[latex]x^2+x - 6 = (x+2)(x-3)[/latex]

Step 2: Apply the Zero-Product Property to find [latex]x[/latex].

[latex]\begin{align*} \text{Equation:} && x^2 + x - 6 &= 0 \\ \text{Factored form:} && (x - 2)(x + 3) &= 0 \\ \text{Apply zero product property:} && x - 2 &= 0 \quad \text{or} \quad x + 3 = 0 \\ \text{Solution:} && x &= 2 \quad \text{or} \quad x = -3  \end{align*}[/latex]

When the leading coefficient is not [latex]1[/latex], we factor a quadratic equation using the method called grouping, which requires four terms.

With the equation in standard form, let’s review the grouping procedures:

  1. With the quadratic in standard form, [latex]ax^2 + bx + c = 0[/latex], multiply [latex]a \cdot c[/latex].
  2. Find two numbers whose product equals [latex]a \cdot c[/latex] and whose sum equals [latex]b[/latex].
  3. Rewrite the equation replacing the [latex]bx[/latex] term with two terms using the numbers found in step 2 as coefficients of [latex]x[/latex].
  4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
  5. Factor out the expression in parentheses.
  6. Set the expressions equal to zero and solve for the variable.
Use grouping to factor and solve the quadratic equation: [latex]4x^2 + 15x + 9 = 0[/latex].

Quadratic Formula

Another method of solving a quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.

quadratic formula

Written in standard form, [latex]a{x}^{2}+bx+c=0[/latex], any quadratic equation can be solved using the quadratic formula:

[latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex]

where [latex]a, b,[/latex] and [latex]c[/latex] are real numbers and [latex]a\ne 0[/latex].

We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by [latex]-1[/latex] and obtain a positive a. Given [latex]a{x}^{2}+bx+c=0[/latex], [latex]a\ne 0[/latex], we will complete the square as follows:

  1. First, move the constant term to the right side of the equal sign:
    [latex]a{x}^{2}+bx=-c[/latex]
  2. As we want the leading coefficient to equal 1, divide through by a:
    [latex]{x}^{2}+\frac{b}{a}x=-\frac{c}{a}[/latex]
  3. Then, find [latex]\frac{1}{2}[/latex] of the middle term, and add [latex]{\left(\frac{1}{2}\frac{b}{a}\right)}^{2}=\frac{{b}^{2}}{4{a}^{2}}[/latex] to both sides of the equal sign:
    [latex]{x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}}=\frac{{b}^{2}}{4{a}^{2}}-\frac{c}{a}[/latex]
  4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:
    [latex]{\left(x+\frac{b}{2a}\right)}^{2}=\frac{{b}^{2}-4ac}{4{a}^{2}}[/latex]
  5. Now, use the square root property, which gives
    [latex]\begin{array}{l}x+\frac{b}{2a}=\pm \sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}\hfill \\ x+\frac{b}{2a}=\frac{\pm \sqrt{{b}^{2}-4ac}}{2a}\hfill \end{array}[/latex]
  6. Finally, add [latex]-\frac{b}{2a}[/latex] to both sides of the equation and combine the terms on the right side. Thus,
    [latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex]
How To: Given a quadratic equation, solve it using the quadratic formula

  1. Make sure the equation is in standard form: [latex]a{x}^{2}+bx+c=0[/latex].
  2. Make note of the values of the coefficients and constant term, [latex]a,b[/latex], and [latex]c[/latex].
  3. Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.
  4. Calculate and solve.
Solve the quadratic equation:

[latex]{x}^{2}+5x+1=0[/latex]