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Solving Quadratic Equations with Square Roots

Solving Quadratic Equations with Square Roots 1.24 - Solution

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a

We use inverse operations to isolate the xx-terms on one side.

x3=5x\dfrac{x}{3}=\dfrac{5}{x}
x=15xx=\dfrac{15}{x}
x2=15x^2=15
x=±15x=\pm \sqrt{15}
Both x=-15x=\text{-} \sqrt{15} and x=15x=\sqrt{15} are solution to the equation.
b

Again, we will use inverse operations in order to get the xx-terms isolated on one side.

3x68x=0\dfrac{3x}{6}-\dfrac{8}{x}=0
3x6=8x\dfrac{3x}{6}=\dfrac{8}{x}
x2=8x\dfrac{x}{2}=\dfrac{8}{x}
x=16xx=\dfrac{16}{x}
x2=16x^2=16
x=±16x=\pm \sqrt{16}
x=±4x=\pm 4

Both x=-4,x=\text{-} 4, and x=4x=4 are solutions to the equation.

c

The left-hand side's least common denominator is 6x.6x. We will use that to rewrite the LHS as a single fraction. After that we will solve the equation using inverse operations.

52x13x=x6\dfrac{5}{2x}-\dfrac{1}{3x}=\dfrac{x}{6}
156x13x=x6\dfrac{15}{6x}-\dfrac{1}{3x}=\dfrac{x}{6}
156x26x=x6\dfrac{15}{6x}-\dfrac{2}{6x}=\dfrac{x}{6}
136x=x6\dfrac{13}{6x}=\dfrac{x}{6}
13=6x2613=\dfrac{6x^2}{6}
13=x213=x^2
x2=13x^2=13
x=±13x=\pm\sqrt{13}

x=-13x=\text{-} \sqrt{13} and x=13x=\sqrt{13} are solutions to the equation.