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Solve the following quadratic equations by factoring: a) 3x(x+5)=0, b) 5x(2x−7)=0, c) 2x2 − 6x = 0, d) 3x2 − 15x = 0, e) 6x2 + 10x = 0.

Mathematics

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Here are the solutions to the problems.

Part a· 2 steps

Solve 3x(x+5)=03x(x+5)=0

  1. Set each factor equal to zero.

    3x=0orx+5=03x = 0 \quad \text{or} \quad x+5 = 0

  2. Solve for xx in each equation.

    x=03orx=−5x = \frac{0}{3} \quad \text{or} \quad x = -5 x=0orx=−5x = 0 \quad \text{or} \quad x = -5

    The solutions are x=0,x=−5\boxed{x=0, x=-5}.

Part b· 2 steps

Solve 5x(2x−7)=05x(2x-7)=0

  1. Set each factor equal to zero.

    5x=0or2x−7=05x = 0 \quad \text{or} \quad 2x-7 = 0

  2. Solve for xx in each equation.

    x=05or2x=7x = \frac{0}{5} \quad \text{or} \quad 2x = 7 x=0orx=72x = 0 \quad \text{or} \quad x = \frac{7}{2}

    The solutions are x=0,x=72\boxed{x=0, x=\frac{7}{2}}.

Part c· 3 steps

Solve 2x2−6x=02x^2 - 6x = 0

  1. Factor out the common term, $2x$.

    2x(x−3)=02x(x-3) = 0

  2. Set each factor equal to zero.

    2x=0orx−3=02x = 0 \quad \text{or} \quad x-3 = 0

  3. Solve for xx in each equation.

    x=02orx=3x = \frac{0}{2} \quad \text{or} \quad x = 3 x=0orx=3x = 0 \quad \text{or} \quad x = 3

    The solutions are x=0,x=3\boxed{x=0, x=3}.

Part d· 3 steps

Solve 3x2−15x=03x^2 - 15x = 0

  1. Factor out the common term, $3x$.

    3x(x−5)=03x(x-5) = 0

  2. Set each factor equal to zero.

    3x=0orx−5=03x = 0 \quad \text{or} \quad x-5 = 0

  3. Solve for xx in each equation.

    x=03orx=5x = \frac{0}{3} \quad \text{or} \quad x = 5 x=0orx=5x = 0 \quad \text{or} \quad x = 5

    The solutions are x=0,x=5\boxed{x=0, x=5}.

Part e· 3 steps

Solve 6x2+10x=06x^2 + 10x = 0

  1. Factor out the common term, $2x$.

    2x(3x+5)=02x(3x+5) = 0

  2. Set each factor equal to zero.

    2x=0or3x+5=02x = 0 \quad \text{or} \quad 3x+5 = 0

  3. Solve for xx in each equation.

    x=02or3x=−5x = \frac{0}{2} \quad \text{or} \quad 3x = -5 x=0orx=−53x = 0 \quad \text{or} \quad x = -\frac{5}{3}

    The solutions are x=0,x=−53\boxed{x=0, x=-\frac{5}{3}}.

Part f· 3 steps

Solve 4x2+20x=04x^2 + 20x = 0

  1. Factor out the common term, $4x$.

    4x(x+5)=04x(x+5) = 0

  2. Set each factor equal to zero.

    4x=0orx+5=04x = 0 \quad \text{or} \quad x+5 = 0

  3. Solve for xx in each equation.

    x=04orx=−5x = \frac{0}{4} \quad \text{or} \quad x = -5 x=0orx=−5x = 0 \quad \text{or} \quad x = -5

    The solutions are x=0,x=−5\boxed{x=0, x=-5}.

Part g· 3 steps

Solve 6x2+2x=06x^2 + 2x = 0

  1. Factor out the common term, $2x$.

    2x(3x+1)=02x(3x+1) = 0

  2. Set each factor equal to zero.

    2x=0or3x+1=02x = 0 \quad \text{or} \quad 3x+1 = 0

  3. Solve for xx in each equation.

    x=02or3x=−1x = \frac{0}{2} \quad \text{or} \quad 3x = -1 x=0orx=−13x = 0 \quad \text{or} \quad x = -\frac{1}{3}

    The solutions are x=0,x=−13\boxed{x=0, x=-\frac{1}{3}}.

Part h· 3 steps

Solve 10x−18x2=010x - 18x^2 = 0

  1. Factor out the common term, $2x$.

    2x(5−9x)=02x(5-9x) = 0

  2. Set each factor equal to zero.

    2x=0or5−9x=02x = 0 \quad \text{or} \quad 5-9x = 0

  3. Solve for xx in each equation.

    x=02or5=9xx = \frac{0}{2} \quad \text{or} \quad 5 = 9x x=0orx=59x = 0 \quad \text{or} \quad x = \frac{5}{9}

    The solutions are x=0,x=59\boxed{x=0, x=\frac{5}{9}}.

Part h(i)· 8 steps

Solve 8x2=6x8x^2 = 6x

  1. Move all terms to one side to set the equation to zero.

    8x2−6x=08x^2 - 6x = 0

  2. Factor out the common term, $2x$.

    2x(4x−3)=02x(4x-3) = 0

  3. Set each factor equal to zero.

    2x=0or4x−3=02x = 0 \quad \text{or} \quad 4x-3 = 0

  4. Solve for xx in each equation.

    x=02or4x=3x = \frac{0}{2} \quad \text{or} \quad 4x = 3 x=0orx=34x = 0 \quad \text{or} \quad x = \frac{3}{4}

    The solutions are x=0,x=34\boxed{x=0, x=\frac{3}{4}}.

    *j) Solve 10x2=15x10x^2 = 15x

  5. Move all terms to one side to set the equation to zero.

    10x2−15x=010x^2 - 15x = 0

  6. Factor out the common term, $5x$.

    5x(2x−3)=05x(2x-3) = 0

  7. Set each factor equal to zero.

    5x=0or2x−3=05x = 0 \quad \text{or} \quad 2x-3 = 0

  8. Solve for xx in each equation.

    x=05or2x=3x = \frac{0}{5} \quad \text{or} \quad 2x = 3 x=0orx=32x = 0 \quad \text{or} \quad x = \frac{3}{2}

    The solutions are x=0,x=32\boxed{x=0, x=\frac{3}{2}}.

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