Answer:
Full explanation is down here
Step-by-step explanation:
Hello!
Recall the standard form of a quadratic: \(ax^2 + bx + c\)
Solve for x:
\(ax^2 + bx + c = 0\)
\(ax^2 + bx = -c\)
\(x^2 + \frac{b}{a}x = \frac{-c}{a}\)
- Complete the square (make sure to add to both sides)
\(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = \frac{-c}{a} + \frac{b^2}{4a^2}\)
- Factor the Perfect Square Trinomial
\((x+\frac{b}{2a})^2 = \frac{-c}{a} + \frac{b^2}{4a^2}\)
- Simplify the right side using a LCD (least common denominator)
\((x+\frac{b}{2a})^2 = \frac{-4ac+b^2}{4a^2}\)
- Take the square root of both sides
\(\sqrt{(x+\frac{b}{2a})^2} = \sqrt{\frac{-4ac+b^2}{4a^2}}\)
- Simplify (make sure to add the plus or minus sign!!)
\(x+\frac{b}{2a} = \frac{\pm\sqrt{-4ac+b^2}}{2a}\)
- Subtract b/2a from both sides
\(x = \frac{\pm\sqrt{-4ac+b^2}}{2a} - \frac{b}{2a}\)
- Convert to traditional format
\(x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Completing the square:
Recall the standard form of a quadratic: \(ax^2 + bx + c\)
To complete a square, we must find the "c" value given \(ax^2 + bx + c\) so that it converts into a perfect square trinomial.
To find c:
- Take the b-value
- Divide it by 2
- Square it
Example: Given the expression \(x^2 - 8x\), complete the square
Let's find the missing "c" value:
- Take the b-value: -8
- Divide it by 2: -4
- Square it: 16
Now, complete the square:
- \(x^2 - 8x\)
- \(x^2 - 8x + 16\)
To quickly factor this into a perfect square trinomial, simply take the value you get from the second step (Divide by 2) and plug it in.
Your factored expression will be (x - 4)²
