we have
[tex]7=-2x^{2} +10x[/tex]
Factor the leading coefficient
[tex]7=-2(x^{2} -5x)[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]7-12.50=-2(x^{2} -5x+2.5^{2})[/tex]
[tex]-5.50=-2(x^{2} -5x+2.5^{2})[/tex]
Divide both sides by [tex]-2[/tex]
[tex]2.75=(x^{2} -5x+2.5^{2})[/tex]
Rewrite as perfect squares
[tex]2.75=(x-2.5)^{2}[/tex]
Taking the square roots of both sides (square root property of equality)
[tex]x-2.5=(+/-)\sqrt{2.75}[/tex]
Remember that
[tex]\sqrt{2.75}=\sqrt{\frac{11}{4}}= \frac{\sqrt{11}}{2}[/tex]
[tex]x-2.5=(+/-)\frac{\sqrt{11}}{2}[/tex]
[tex]x=2.5(+/-)\frac{\sqrt{11}}{2}[/tex]
[tex]x=2.5+\frac{\sqrt{11}}{2}=\frac{5+\sqrt{11}}{2}[/tex]
[tex]x=2.5-\frac{\sqrt{11}}{2}=\frac{5-\sqrt{11}}{2}[/tex]
the answer is
The solutions are
[tex]x=\frac{5+\sqrt{11}}{2}[/tex]
[tex]x=\frac{5-\sqrt{11}}{2}[/tex]
