Answer:
The solutions are:
[tex]x=\frac{7-\sqrt{17}}{2},\:x=\frac{7+\sqrt{17}}{2}[/tex]
Step-by-step explanation:
Given the quadratic equation
-x² + 7x = 8
Solving using the quadratic formula
[tex]-x^2+7x=8[/tex]
Subtract 8 from both sides
[tex]-x^2+7x-8=8-8[/tex]
Simplify
[tex]-x^2+7x-8=0[/tex]
For a quadratic function of the form ax² + bx + c = 0, the solutions are:
[tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
For a = -1, b = 7, c = -8
[tex]x_{1,\:2}=\frac{-7\pm \sqrt{7^2-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}[/tex]
[tex]x_{1,\:2}=\frac{-7\pm \:\sqrt{49-32}}{2\left(-1\right)}[/tex]
[tex]x_{1,\:2}=\frac{-7\pm \sqrt{17}}{2\left(-1\right)}[/tex]
Separate the solutions
[tex]x_1=\frac{-7+\sqrt{17}}{2\left(-1\right)},\:x_2=\frac{-7-\sqrt{17}}{2\left(-1\right)}v[/tex]
solving
[tex]x_1=\frac{-7+\sqrt{17}}{2\left(-1\right)}[/tex]
[tex]=\frac{-7+\sqrt{17}}{-2}[/tex]
Apply the fraction rule: [tex]\frac{-a}{-b}=\frac{a}{b}[/tex]
i.e. [tex]-7+\sqrt{17}=-\left(7-\sqrt{17}\right)[/tex]
so
[tex]=\frac{7-\sqrt{17}}{2}[/tex]
Similarly solving
[tex]x_2=\frac{-7-\sqrt{17}}{2\left(-1\right)}[/tex]
[tex]=\frac{-7-\sqrt{17}}{-2\cdot \:1}[/tex]
Apply the fraction rule: [tex]\frac{-a}{-b}=\frac{a}{b}[/tex]
i.e. [tex]-7-\sqrt{17}=-\left(7+ \sqrt{17}\right)[/tex]
[tex]=\frac{7+\sqrt{17}}{2}[/tex]
Therefore, the solutions are:
[tex]x=\frac{7-\sqrt{17}}{2},\:x=\frac{7+\sqrt{17}}{2}[/tex]
