Respuesta :
[tex](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}}[/tex]
The solutions of the equation [tex]x^{2}+8x-3=0[/tex] are [tex]\boxed{0.36\text{ and }-8.36}[/tex].
Further explanation:
Given:
The equation is [tex]x^{2}+8x-3=0[/tex].
Concept used:
A quadratic function is defined as a function with degree or the highest power of the variable as [tex]2[/tex].
The standard form of the quadratic equation is,
[tex]\boxed{ax^{2}+bx+c=0}[/tex] … (1)
The above equation is converted by completing square as follows:
[tex]\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}[/tex]
Calculation:
The given equation is [tex]x^{2}+8x-3=0[/tex].
Now, add and subtract by [tex]16[/tex] in the left hand side of the equation [tex]x^{2}+8x-3=0[/tex] and simplify as follows,
[tex]\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}[/tex]
Further simplify the above equation as follows,
[tex]\begin{aligned}(x+4)^{2}&=19\\x+4&=\pm \sqrt{19}\\x+4&=\pm 4.36\\x&=\pm 4.367-4\end{aligned}[/tex]
First take positive sign and obtain the value of [tex]x[/tex] as follows:
[tex]\begin{aligned}x&=4.36-4\\&=0.36\end{aligned}[/tex]
Now, take negative sign and obtain the value of [tex]x[/tex] as follows:
[tex]\begin{aligned}x&=-4.36-4\\&=-8.36\end{aligned}[/tex]
The values of [tex]x[/tex] are [tex]0.36\text{ and }-8.36[/tex].
Therefore, the solutions of the given equation are [tex]0.36\text{ and }-8.36[/tex].
To check that [tex]0.36[/tex] is the solution of the equation, substitute [tex]x=0.36[/tex] in the equation [tex]x^{2}+8x-3=0[/tex] as,
[tex]\begin{aligned}(0.36)^{2}+(8\times 0.36)-3\ _{=}^{?}\ 0\\0.1296+2.88-3\ _{=}^{?}\ 0\\0=0\end{aligned}[/tex]
Therefore, [tex]0.36[/tex] is the solution of the equation [tex]x^{2}+8x-3=0[/tex].
To check that [tex]-8.36[/tex] is the solution of the equation, substitute [tex]x=-8.36[/tex] in the equation [tex]x^{2}+8x-3=0[/tex] as follows:
[tex]\begin{aligned}(-8.36)^{2}+(8\times (-8.36))-3\ _{=}^{?}\ 0\\69.8896-66.88-3\ _{=}^{?}\ 0\\0=0\end{aligned}[/tex]
Therefore, [tex]-8.36[/tex] is the solution of the equation [tex]x^{2}+8x-3=0[/tex].
Learn more:
1. A problem on quadratic function https://brainly.com/question/1332667
2. A problem on parabola https://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.
