Answer
\( y^{2} - 9x^{2} = (y - 3x)(y + 3x) \)
Solution
Solve the equation \( x(3x+13)=11 \).
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
\(x\left(3x+13\right)=11\)
- step1: Expand the expression:
\(3x^{2}+13x=11\)
- step2: Move the expression to the left side:
\(3x^{2}+13x-11=0\)
- step3: Solve using the quadratic formula:
\(x=\frac{-13\pm \sqrt{13^{2}-4\times 3\left(-11\right)}}{2\times 3}\)
- step4: Simplify the expression:
\(x=\frac{-13\pm \sqrt{13^{2}-4\times 3\left(-11\right)}}{6}\)
- step5: Simplify the expression:
\(x=\frac{-13\pm \sqrt{301}}{6}\)
- step6: Separate into possible cases:
\(\begin{align}&x=\frac{-13+\sqrt{301}}{6}\\&x=\frac{-13-\sqrt{301}}{6}\end{align}\)
- step7: Rewrite the fraction:
\(\begin{align}&x=\frac{-13+\sqrt{301}}{6}\\&x=-\frac{13+\sqrt{301}}{6}\end{align}\)
- step8: Rewrite:
\(x_{1}=-\frac{13+\sqrt{301}}{6},x_{2}=\frac{-13+\sqrt{301}}{6}\)
The equation provided is not in a format that can be solved directly. If you could provide the equation in a standard format, I would be happy to assist you further.
Unable to solve the equation.
Solve the equation \( \sqrt{(x-2)^{-3}}=64 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\sqrt{\left(x-2\right)^{-3}}=64\)
- step1: Find the domain:
\(\sqrt{\left(x-2\right)^{-3}}=64,x>2\)
- step2: Simplify the root:
\(\frac{1}{\sqrt{\left(x-2\right)^{3}}}=64\)
- step3: Cross multiply:
\(1=\sqrt{\left(x-2\right)^{3}}\times 64\)
- step4: Simplify the equation:
\(1=64\sqrt{\left(x-2\right)^{3}}\)
- step5: Swap the sides:
\(64\sqrt{\left(x-2\right)^{3}}=1\)
- step6: Divide both sides:
\(\frac{64\sqrt{\left(x-2\right)^{3}}}{64}=\frac{1}{64}\)
- step7: Divide the numbers:
\(\sqrt{\left(x-2\right)^{3}}=\frac{1}{64}\)
- step8: Raise both sides to the \(2\)-th power\(:\)
\(\left(\sqrt{\left(x-2\right)^{3}}\right)^{2}=\left(\frac{1}{64}\right)^{2}\)
- step9: Evaluate the power:
\(\left(x-2\right)^{3}=\frac{1}{4096}\)
- step10: Simplify the equation:
\(\sqrt[3]{\left(x-2\right)^{3}}=\sqrt[3]{\frac{1}{4096}}\)
- step11: Calculate:
\(x-2=\sqrt[3]{\frac{1}{4096}}\)
- step12: Simplify the root:
\(x-2=\frac{1}{16}\)
- step13: Move the constant to the right side:
\(x=\frac{1}{16}+2\)
- step14: Add the numbers:
\(x=\frac{33}{16}\)
- step15: Check if the solution is in the defined range:
\(x=\frac{33}{16},x>2\)
- step16: Find the intersection:
\(x=\frac{33}{16}\)
- step17: Check the solution:
\(x=\frac{33}{16}\)
The solution to the equation \( \sqrt{(x-2)^{-3}}=64 \) is \( x = \frac{33}{16} \) or \( x = 2.0625 \).
\( x = \frac{33}{16} \) or \( x = 2.0625 \)
Factor the expression \( y^{2}-9x^{2} \).
Factor the expression by following steps:
- step0: Factor:
\(y^{2}-9x^{2}\)
- step1: Rewrite the expression:
\(y^{2}-\left(3x\right)^{2}\)
- step2: Factor the expression:
\(\left(y-3x\right)\left(y+3x\right)\)
The factored form of the expression \( y^{2}-9x^{2} \) is \( (y-3x)(y+3x) \).
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