Complete the Square
Learning Outcome
- Complete the square to create a perfect square trinomial
- Given a quadratic equation that cannot be factored and with [latex]a=1[/latex], first add or subtract the constant term to the right sign of the equal sign.
[latex]{x}^{2}+4x=-1[/latex]
- Multiply the b term by [latex]\frac{1}{2}[/latex] and square it.
[latex]\begin{array}{l}\frac{1}{2}\left(4\right)=2\hfill \\ {2}^{2}=4\hfill \end{array}[/latex]
- Add [latex]{\left(\frac{1}{2}b\right)}^{2}[/latex] to both sides of the equal sign and simplify the right side. We have
[latex]\begin{array}{l}{x}^{2}+4x+4=-1+4\hfill \\ {x}^{2}+4x+4=3\hfill \end{array}[/latex]
- The left side of the equation can now be factored as a perfect square.
[latex]\begin{array}{l}{x}^{2}+4x+4=3\hfill \\ {\left(x+2\right)}^{2}=3\hfill \end{array}[/latex]
- Use the square root property and solve.
[latex]\begin{array}{l}\sqrt{{\left(x+2\right)}^{2}}=\pm \sqrt{3}\hfill \\ x+2=\pm \sqrt{3}\hfill \\ x=-2\pm \sqrt{3}\hfill \end{array}[/latex]
- The solutions are [latex]x=-2+\sqrt{3}[/latex], [latex]x=-2-\sqrt{3}[/latex].
How To: use the method of complete the square to write a perfect square trinomial from an expression.
- Given an expression of the form [latex]a\left(x^2+bx\right)[/latex], add [latex]\left(\dfrac{b}{2}\right)^2[/latex] inside the parentheses.
- Then subtract [latex]a\left(\dfrac{b}{2}\right)^2[/latex] to counteract the change you made to the expression.
- If completing the square on one side of an equation, you may either subtract the value of [latex]a\left(\dfrac{b}{2}\right)^2[/latex] from that side, or add it to the other to maintain equality.
- Then factor the perfect square trinomial you created inside the original parentheses.
The resulting form will look like this:
Given[latex]\qquad a\left(x^2+bx\right)[/latex]
add [latex]\left(b/2\right)^2[/latex] inside the parentheses and subtract [latex]a\left(b/2\right)^2[/latex] to counteract the change you made to the expression[latex]=a\left(x^2+bx+ \left(\dfrac{b}{2}\right)^2\right)-a\left(\dfrac{b}{2}\right)^2[/latex]
then factor the resulting perfect square trinomial[latex]=a\left(x+ \dfrac{b}{2}\right)^2-a\left(\dfrac{b}{2}\right)^2[/latex].
Example : Create a perfect square trinomial using the method of complete the square
Complete the square on: [latex]3\left(x^2 - 10x\right)[/latex].Answer: Add [latex]\left(\dfrac{b}{2}\right)^2[/latex] inside the parentheses and subtract [latex]a\left(\dfrac{b}{2}\right)^2[/latex]
[latex]3\left(x^2 - 10x+25\right)-3\cdot25[/latex]
Factor the perfect square trinomial and simplify[latex]3\left(x -5\right)^2-75[/latex].
.Analysis
The resulting expression is equivalent to the original expression. To test this, substitute a small value for [latex]x[/latex], say [latex]x=3[/latex]. [latex]3\left(3^2-10\cdot3\right) = \quad -63 \quad = 3(3-5)^2-75[/latex]. True.try it
[ohm_question]15002[/ohm_question]Licenses & Attributions
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