Question

What's 5 - the square root of x + 10 = the square root of 7 - x

Answer 1

`\bf 5-√(x+10)=√(7-x)\impliedby \textit{squaring both sides} \\\\\\ (5-√(x+10))^2=(√(7-x))^2 \\\\\\ (5^2)-10√(x+10)+(x+10)=7-x \\\\\\ 35-10√(x+10)+x=7-x\implies 28-10√(x+10)+2x=0 \\\\\\ 2x+28=10√(x+10)\implies x+14=5√(x+10)\impliedby \begin{array}{llll} squaring\ both\\ sides\ again \end{array} \\\\\\`


`\bf (x^2)+28x+(14^2)=5^2(x+10)\implies x^2+28x+196=25x+250 \\\\\\ \begin{array}{lcclll} x^2&+3x&-54&=0\\ &\uparrow &\uparrow \\ &-6+9&-6\cdot 9 \end{array} \implies (x+9)(x-6)=0\implies \begin{cases} x+9=0\\ \boxed{x=-9}\\ -----\\ x-6=0\\ \boxed{x=6} \end{cases}`