Question

what is the value of x?
$\sqrt{x - 3} + \sqrt{3x + 4} = 5$
​

Answer 1

Answer:

$\green{\sqrt{x-3}+\sqrt{3x+4}=5\quad :\quad x=4}$

Step-by-step explanation:

$\sqrt{x-3}+\sqrt{3x+4}=5}$

$\gray{\mathrm{Subtract\:}\sqrt{3x+4}\mathrm{\:from\:both\:sides}}$

$\sqrt{x-3}+\sqrt{3x+4}-\sqrt{3x+4}=5-\sqrt{3x+4}$

$\gray{\mathrm{Simplify}}$

$\sqrt{x-3}=5-\sqrt{3x+4}$

$\gray{\mathrm{Square\:both\:sides}}$

$\left(\sqrt{x-3}\right)^2=\left(5-\sqrt{3x+4}\right)^2$

$\gray{\mathrm{Expand\:}\left(\sqrt{x-3}\right)^2:\quad x-3}$

$\gray{\mathrm{Expand\:}\left(5-\sqrt{3x+4}\right)^2:\quad 3x+29-10\sqrt{3x+4}}$

$x-3=3x+29-10\sqrt{3x+4}$

$\gray{\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}}$

$x-3-3x=3x+29-10\sqrt{3x+4}-3x$

$\gray{\mathrm{Simplify}}$

$-2x-3=-10\sqrt{3x+4}+29$

$\gray{\mathrm{Subtract\:}29\mathrm{\:from\:both\:sides}}$

$-2x-3-29=-10\sqrt{3x+4}+29-29$

$\gray{\mathrm{Simplify}}$

$-2x-32=-10\sqrt{3x+4}$

$\gray{\mathrm{Square\:both\:sides}}$

$\left(-2x-32\right)^2=\left(-10\sqrt{3x+4}\right)^2$

$\gray{\mathrm{Expand\:}\left(-2x-32\right)^2:\quad 4x^2+128x+1024}$

$\gray{\mathrm{Expand\:}\left(-10\sqrt{3x+4}\right)^2:\quad 300x+400}$

$4x^2+128x+1024=300x+400$

$\black{\mathrm{Solve\:}\:4x^2+128x+1024=300x+400:}$

$4x^2+128x+1024=300x+400$

$\gray{\mathrm{Subtract\:}400\mathrm{\:from\:both\:sides}}$

$4x^2+128x+1024-400=300x+400-400$

$\gray{\mathrm{Simplify}}$

$4x^2+128x+624=300x$

$\gray{\mathrm{Subtract\:}300x\mathrm{\:from\:both\:sides}}$

$4x^2+128x+624-300x=300x-300x$

$\gray{\mathrm{Simplify}}$

$4x^2-172x+624=0$

$\gray{\mathrm{Solve\:with\:the\:quadratic\:formula}}$

$\blue{\mathrm{Quadratic\:Equation\:Formula:}}$

$\bullet~~\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}$ $x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\gray{\mathrm{For\:}\quad a=4,\:b=-172,\:c=624:\quad x_{1,\:2}=\frac{-\left(-172\right)\pm \sqrt{\left(-172\right)^2-4\cdot \:4\cdot \:624}}{2\cdot \:4}}$

$x=\frac{-\left(-172\right)+\sqrt{\left(-172\right)^2-4\cdot \:4\cdot \:624}}{2\cdot \:4}:\quad 39$

$x=\frac{-\left(-172\right)-\sqrt{\left(-172\right)^2-4\cdot \:4\cdot \:624}}{2\cdot \:4}:\quad 4$

$\gray{\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}}$

$x=39,\:x=4$

$\black{\mathrm{Verify\:Solutions}:}$

$\gray{\mathrm{Check\:the\:solutions\:by\:plugging\:them\:into\:}\sqrt{x-3}+\sqrt{3x+4}=5}$ $\gray{\mathrm{Remove\:the\:ones\:that\:don't\:agree\:with\:the\:equation.}}$

$\black{\mathrm{Plug}\:x=39:}$

$\sqrt{39-3}+\sqrt{3\cdot \:39+4}=5$

$\gray{\sqrt{39-3}+\sqrt{3\cdot \:39+4}=17}$

$\gray{17=5}$

$\red{\mathrm{False}}$

$\black{\mathrm{Plug}\:x=4:}$

$\sqrt{4-3}+\sqrt{3\cdot \:4+4}=5$

$\gray{\sqrt{4-3}+\sqrt{3\cdot \:4+4}=5}$

$\gray{5=5}$

$\green{\mathrm{True}}$

$\gray{\mathrm{The\:solution\:is}}$

$x=4$