Question

Find all real numbers x that satisfy the equation \sqrt{x^2-2}=\sqrt{3-x}

177

likes

884 views

Answer to a math question Find all real numbers x that satisfy the equation \sqrt{x^2-2}=\sqrt{3-x}

$Expert avatar$

Murray

4.5

90 Answers

${x}^{2}-2=3-x$

${x}^{2}-2-3+x=0$

${x}^{2}-5+x=0$

${x}^{2}+x-5=0$

$\begin{array} { l }a=1,& b=1,& c=-5\end{array}$

$x=\frac{ -1\pm\sqrt{ {1}^{2}-4 \times 1 \times \left( -5 \right) } }{ 2 \times 1 }$

$x=\frac{ -1\pm\sqrt{ {1}^{2}-4 \times \left( -5 \right) } }{ 2 \times 1 }$

$x=\frac{ -1\pm\sqrt{ {1}^{2}-4 \times \left( -5 \right) } }{ 2 }$

$x=\frac{ -1\pm\sqrt{ 1-4 \times \left( -5 \right) } }{ 2 }$

$x=\frac{ -1\pm\sqrt{ 1+20 } }{ 2 }$

$x=\frac{ -1\pm\sqrt{ 21 } }{ 2 }$

$\begin{array} { l }x=\frac{ -1+\sqrt{ 21 } }{ 2 },\\x=\frac{ -1-\sqrt{ 21 } }{ 2 }\end{array}$

$\begin{array} { l }\sqrt{ {\left( \frac{ -1+\sqrt{ 21 } }{ 2 } \right)}^{2}-2 }=\sqrt{ 3-\frac{ -1+\sqrt{ 21 } }{ 2 } },\\x=\frac{ -1-\sqrt{ 21 } }{ 2 }\end{array}$

$\begin{array} { l }\sqrt{ {\left( \frac{ -1+\sqrt{ 21 } }{ 2 } \right)}^{2}-2 }=\sqrt{ 3-\frac{ -1+\sqrt{ 21 } }{ 2 } },\\\sqrt{ {\left( \frac{ -1-\sqrt{ 21 } }{ 2 } \right)}^{2}-2 }=\sqrt{ 3-\frac{ -1-\sqrt{ 21 } }{ 2 } }\end{array}$

$\begin{array} { l }\frac{ \sqrt{ 14-2\sqrt{ 21 } } }{ 2 }=\frac{ \sqrt{ 14-2\sqrt{ 21 } } }{ 2 },\\\sqrt{ {\left( \frac{ -1-\sqrt{ 21 } }{ 2 } \right)}^{2}-2 }=\sqrt{ 3-\frac{ -1-\sqrt{ 21 } }{ 2 } }\end{array}$

$\begin{array} { l }\frac{ \sqrt{ 14-2\sqrt{ 21 } } }{ 2 }=\frac{ \sqrt{ 14-2\sqrt{ 21 } } }{ 2 },\\\frac{ \sqrt{ 14+2\sqrt{ 21 } } }{ 2 }=\frac{ \sqrt{ 14+2\sqrt{ 21 } } }{ 2 }\end{array}$

$\begin{array} { l }x=\frac{ -1+\sqrt{ 21 } }{ 2 },\\\frac{ \sqrt{ 14+2\sqrt{ 21 } } }{ 2 }=\frac{ \sqrt{ 14+2\sqrt{ 21 } } }{ 2 }\end{array}$

$\begin{array} { l }x=\frac{ -1+\sqrt{ 21 } }{ 2 },\\x=\frac{ -1-\sqrt{ 21 } }{ 2 }\end{array}$

$\begin{align*}&\begin{array} { l }x_1=\frac{ -1-\sqrt{ 21 } }{ 2 },& x_2=\frac{ -1+\sqrt{ 21 } }{ 2 }\end{array} \\&\begin{array} { l }x_1\approx-2.79129,& x_2\approx1.79129\end{array}\end{align*}$

Frequently asked questions (FAQs)

Question: What is the area, in square units, of a rectangle with length 8 units and width 12 units?

What is the measure of an angle formed by a tangent line and a radius in a circle?

What is the result of multiplying 25 by 8 and then adding 17?

New questions in Mathematics

1/2x +3 <4x-7

If O(3,-2) is reflected across x = 2. What are the coordinates of O

X^2 = 25

Determine the equations of the recipes that pass through the following pairs of points P1 (2;-1) and p2 (4;-1)

[(36,000,000)(0.000003)^2]divided(0.00000006)

The data set (75, 85, 58, 72, 70, 75) is a random sample from the normal distribution No(µ, σ). Determine a 95% two-sided confidence interval for the mean µ .

prove that if n odd integer then n^2+5 is even