Presentation on theme: "Why we complete the square We have learned how to factor quadratic expressions to solve. Many quadratic equations contain expressions that cannot be."— Presentation transcript: 1 Why we complete the square We have learned how to factor quadratic
expressions to solve. Many quadratic equations contain expressions that cannot be easily factored. For equations containing these types of expressions, you can use square roots to find roots. 2 If a quadratic expression of the
form x 2 + bx + c is not a perfect square trinomial, you can add a term to form a perfect square trinomial. This is called completing the square. 3 Assignment #41 Pg. 237, #13-17odd, 18, 28-33
4 DO NOW Get out your laptop and a pencil. 5 You can complete the square to solve quadratic equations. Completing the Square
6 Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 4x + Add. Factor. Find. x 2 + 4x + 4 (x + 2) 2 Check Find the square of the binomial. = x 2 + 4x + 4 (x + 2) 2 = (x + 2)(x + 2) Completing The Square: 1
7 Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 3x + Add. Factor. Find. Check Find the square of the binomial. Completing The Square: 2
8 Solve the equation by completing the square. x 2 = 12x – 20 x 2 – 12x = –20 Collect variable terms on one side. Simplify. Set up to complete the square. x 2 – 12x + = –20 + Add to both sides. x 2 – 12x + 36 = –20 + 36 Completing The Square To Find Roots: 1
9 (x – 6) 2 = 16 Factor. Take the square root of both sides. x – 6 = ±4 Solve for x. x – 6 = 4 or x – 6 = –4 Simplify. x = 10 or x = 2 Completing The Square To Find Roots: 1
10 Vertex Form The result of completing the square and factoring was: (x – 6) 2 = 16 If we subtract 16 from both sides, we end up with our function in vertex form. f(x) = (x – 6) 2 - 16
11 Solve the equation by completing the square. 18x + 3x 2 = 45 x 2 + 6x = 15 Divide both sides by 3. Simplify. x 2 + 6x + = 15 + Add to both sides. x 2 + 6x + 9 = 15 + 9 Set up to complete the square. Completing The Square To Find Roots: 2
12 Take the square root of both sides. Factor. Simplify. (x + 3) 2 = 24 Completing The Square To Find Roots: 2 13 3x 2 – 24x = 27 Solve the equation
by completing the square. Divide both sides by 3. Simplify. Add to both sides. Set up to complete the square. x 2 – 8x = 9 x 2 –8x + = 9 + Completing The Square To Find Roots: 3 14 Solve the equation by completing the square. Factor. Solve for x. Simplify. Take the square root of both sides. x =–1 or x = 9 x – 4 =–5
or x – 4 = 5 Completing The Square To Find Roots: 3 15 Assignment #42 Pg. 237, #19-27odd, 40-44even, 52 If you've got a quadratic equation on the form of $$ax^{2}+c=0$$ Then you can solve the equation by using the square root of $$x=\pm \sqrt{\frac{-c}{a}}$$ Example $$3x^{2}-243=0$$ $$3x^{2}=243$$ $$x^{2}=\frac{243}{3}$$ $$x^{2}=81$$ $$x=\pm \sqrt{81}$$ $$x=9\: \: or\: \: x=-9$$ This method can only be used if b = 0. If we instead have an equation on the form of $$x^{2}+bx=0$$ we can't use the square root initially since we do not have c-value. But we can add a constant d to both sides of the equation to get a new equivalent equation that is a perfect square trinomial. Remember that a perfect square trinomial can be written as $$x^{2}+bx + d=\left ( x+d \right )^{2}=0$$ This process is called completing the square and the constant d we're adding is $$d=\left (\frac{b}{2} \right )^{2}$$ Example $$x^{2}+12x=0$$ We begin by finding the constant d that can be used to complete the square. $$d=\left (\frac{b}{2} \right )^{2}=\left ( \frac{12}{2} \right )^{2}=6^{2}=36$$ $$x^{2}+12x+d=0+d\Rightarrow$$ $$x^{2}+12x+36=0+36\Rightarrow$$ $$\begin{pmatrix}x+6 \end{pmatrix}^{2}=36$$ $$\sqrt{\begin{pmatrix} x+6 \end{pmatrix}^{2}}=\pm \sqrt{36}$$ $$\begin{matrix} x+6=6\: \: &or\: \: & x+6=-6\\ x=0 & & x=-12 \end{matrix}$$ The completing the square method could of course be used to solve quadratic equations on the form of $$ax^{2}+bx+c=0$$ In this case you will add a constant d that satisfy the formula $$d=\left ( \frac{b}{2} \right )^{2}-c$$ Video lessonSolve the equation by completing the squares $$x^{2} - 3x - 10=0$$ |