Question

Solve x2 + 8x − 3 = 0 using the completing-the-square method.

Answer 1

The solutions of the equation $x^{2}+8x-3=0$ are $\boxed{0.36\text{ and }-8.36}$.

Further explanation:

Given:

The equation is $x^{2}+8x-3=0$.

Concept used:

A quadratic function is defined as a function with degree or the highest power of the variable as $2$.

The standard form of the quadratic equation is,

$\boxed{ax^{2}+bx+c=0}$                     … (1)

The above equation is converted by completing square as follows:

$\begin{aligned}x^{2}+bx+c+\left(\dfrac{b}{2}\right)^{2}-\left(\dfrac{b}{2}\right)^{2}&=0\\x^{2}+bx+\left(\dfrac{b}{2}\right)^{2}+c-\left(\dfrac{b}{2}\right)^{2}&=0\\ \left(x+\dfrac{b}{2}\right)^{2}+c-\left(\dfrac{b}{2}\right)^{2}&=0\end{aligned}$  

Calculation:

The given equation is $x^{2}+8x-3=0$.

Now, add and subtract by $16$ in the left hand side of the equation $x^{2}+8x-3=0$ and simplify as follows,

$\begin{aligned}x^{2}+8x-3+16-16&=0\\ (x^{2}+8x+16)-3-16&=0\\ (x+4)^{2}-19&=0\\(x+4)^{2}&=19\end{aligned}$  

Further simplify the above equation as follows,

$\begin{aligned}(x+4)^{2}&=19\\x+4&=\pm \sqrt{19}\\x+4&=\pm 4.36\\x&=\pm 4.367-4\end{aligned}$  

First take positive sign and obtain the value of $x$ as follows:

$\begin{aligned}x&=4.36-4\\&=0.36\end{aligned}$  

Now, take negative sign and obtain the value of $x$ as follows:

$\begin{aligned}x&=-4.36-4\\&=-8.36\end{aligned}$

The values of $x$ are $0.36\text{ and }-8.36$.

Therefore, the solutions of the given equation are $0.36\text{ and }-8.36$.  

To check that $0.36$ is the solution of the equation, substitute $x=0.36$ in the equation $x^{2}+8x-3=0$ as,

$\begin{aligned}(0.36)^{2}+(8\times 0.36)-3\ _{=}^{?}\ 0\\0.1296+2.88-3\ _{=}^{?}\ 0\\0=0\end{aligned}$  

Therefore, $0.36$ is the solution of the equation $x^{2}+8x-3=0$.

To check that $-8.36$ is the solution of the equation, substitute $x=-8.36$ in the equation $x^{2}+8x-3=0$ as follows:

$\begin{aligned}(-8.36)^{2}+(8\times (-8.36))-3\ _{=}^{?}\ 0\\69.8896-66.88-3\ _{=}^{?}\ 0\\0=0\end{aligned}$

Therefore, $-8.36$ is the solution of the equation $x^{2}+8x-3=0$.

Learn more:

1. A problem on quadratic function brainly.com/question/1332667

2. A problem on parabola brainly.com/question/3213890

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Quadratic equation

Keywords: Quadratic equation, complete square, function, equations, degree, highest power, solution, standard form, x2+8x-3, variable.

Answer 2

$(a+b)^2=a^2+2ab+b^2\\------------------\\\\x^2+8x-3=0\ \ \ |add\ 3\ to\ both\ sides\\\\x^2+2x\cdot4=3\ \ \ |add\ 4^2\ to\ both\ sides\\\\x^2+2x\cdot4+4^2=3+4^2\\\\(x+4)^2=3+16\\\\(x+4)^2=19\iff x+4=-\sqrt{19}\ or\ x+4=\sqrt{19}\\|subtract\ 4\ from\ both\ sides\\\\\boxed{x=-4-\sqrt{19}\ or\ x=-4+\sqrt{19}}$