# How do you solve by completing the square: x^2 – 4x – 60 = 0?

Apr 3, 2015

Solving a quadratic expression by completing the square means to manipulate the expression in order to write it in the form
${\left(x + a\right)}^{2} = b$
So, if $b \setminus \ge 0$, you can take the square root at both sides to get
$x + a = \setminus \pm \setminus \sqrt{b}$
and conclude $x = \setminus \pm \setminus \sqrt{b} - a$.

Now, we have ${\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2}$. Since you equation starts with ${x}^{2} - 4 x$, this means that $2 a x = - 4 x$, and so $a = - 2$.
Adding $64$ at both sides, we have
${x}^{2} - 4 x + 4 = 64$
Which is the form we wanted, because now we have
${\left(x - 2\right)}^{2} = 64$
$x - 2 = \setminus \pm \setminus \sqrt{64} = \setminus \pm 8$ and finally $x = \setminus \pm 8 + 2$, which means that the two solutions are $- 8 + 2 = - 6$ and $8 + 2 = 10$

Apr 3, 2015
• First, we Transpose the Constant to one side of the equation.
Transposing $- 60$ to the other side we get:
${x}^{2} - 4 x = 60$

• Application of ${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$
We look at the Co-efficient of $x$. It's $- 4$
We take half of this number (including the sign), giving us –2
We square this value to get ${\left(- 2\right)}^{2} = 4$. We add this number to BOTH sides of the Equation.
${x}^{2} - 4 x + 4 = 60 + 4$
${x}^{2} - 4 x + 4 = 64$
The Left Hand side ${x}^{2} - 4 x + 4$ is in the form ${a}^{2} - 2 a b + {b}^{2}$
where $a$ is $x$, and $b$ is $2$

• The equation can be written as
${\left(x - 2\right)}^{2} = 64$

So $\left(x - 2\right)$ can take either $8$ or $- 8$ as a value. That's because squaring both will give us 64.

$x - 2 = 8$ (or) $x - 2 = - 8$
$x = 10$ (or) $x = - 6$

• Solution : $x = 10 , - 6$

• Verify your answer by substituting these values in the Original Equation ${x}^{2} - 4 x - 60 = 0$
The Left hand Side is ${x}^{2} - 4 x - 60$, and the Right Hand Side is $0$

If x = 10,
Left Hand Side
$= {\left(10\right)}^{2} - 4 \left(10\right) - 60$
$= 100 - 40 - 60$
$= 0$ (Is Equal to the Right Hand Side)

If x = -6,
Left Hand Side
$= {\left(- 6\right)}^{2} - 4 \left(- 6\right) - 60$
$= 36 + 24 - 60$
$= 0$ (Is Equal to the Right Hand Side)

Both the solutions are verified. Our solution $x = 10 , - 6$ is correct.