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Solving Quadratic Equations with Square Roots

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Algebra 1
Quadratic Equations
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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}
MultEqnLHS⋅3=RHS⋅3\text{LHS} \cdot 3=\text{RHS}\cdot 3
x=15xx=\dfrac{15}{x}
MultEqnLHS⋅x=RHS⋅x\text{LHS} \cdot x=\text{RHS}\cdot x
x2=15x^2=15
SqrtEqnLHS=RHS\sqrt{\text{LHS}}=\sqrt{\text{RHS}}
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.

3x6−8x=0\dfrac{3x}{6}-\dfrac{8}{x}=0
AddEqnLHS+8x=RHS+8x\text{LHS}+\dfrac{8}{x}=\text{RHS}+\dfrac{8}{x}
3x6=8x\dfrac{3x}{6}=\dfrac{8}{x}
SimpQuotSimplify quotient
x2=8x\dfrac{x}{2}=\dfrac{8}{x}
MultEqnLHS⋅2=RHS⋅2\text{LHS} \cdot 2=\text{RHS}\cdot 2
x=16xx=\dfrac{16}{x}
MultEqnLHS⋅x=RHS⋅x\text{LHS} \cdot x=\text{RHS}\cdot x
x2=16x^2=16
SqrtEqnLHS=RHS\sqrt{\text{LHS}}=\sqrt{\text{RHS}}
x=±16x=\pm \sqrt{16}
SimpRootSimplify root
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.

52x−13x=x6\dfrac{5}{2x}-\dfrac{1}{3x}=\dfrac{x}{6}
ExpandFracab=a⋅3b⋅3\dfrac{a}{b}=\dfrac{a \cdot 3}{b \cdot 3}
156x−13x=x6\dfrac{15}{6x}-\dfrac{1}{3x}=\dfrac{x}{6}
ExpandFracab=a⋅2b⋅2\dfrac{a}{b}=\dfrac{a \cdot 2}{b \cdot 2}
156x−26x=x6\dfrac{15}{6x}-\dfrac{2}{6x}=\dfrac{x}{6}
SubFracSubtract fractions
136x=x6\dfrac{13}{6x}=\dfrac{x}{6}
MultEqnLHS⋅6x=RHS⋅6x\text{LHS} \cdot 6x=\text{RHS}\cdot 6x
13=6x2613=\dfrac{6x^2}{6}
SimpQuotSimplify quotient
13=x213=x^2
RearrangeEqnRearrange equation
x2=13x^2=13
SqrtEqnLHS=RHS\sqrt{\text{LHS}}=\sqrt{\text{RHS}}
x=±13x=\pm\sqrt{13}

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