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Victor is y years old. His brother Fred is four years older than Victor. The product of their ages is 780. a) Set up an equation. b) Find Victor's age.

Mathematics

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Answers

2 parts
a

y2+4y=780y^2 + 4y = 780

b

26 years old26 \text{ years old}

Step-by-step solution

8 steps

Here is the solution to :

21) Victor is yy years old. His brother Fred is four years older than Victor. The product of their ages is 780.

Part a· 3 steps

Set up an equation to represent this information.

  1. Define the ages in terms of yy.

    Victor's age = yy
    Fred's age = y+4y+4

  2. Write the equation for the product of their ages.

    y(y+4)=780y(y+4) = 780

  3. Expand the equation.

    y2+4y=780y^2 + 4y = 780

    The equation is:

    y2+4y=780\boxed{y^2 + 4y = 780}

Part b

26 years old26 \text{ years old}

Solve your equation from (a) to find Victor's age.

  1. Rearrange the equation into a standard quadratic form.

    y2+4y−780=0y^2 + 4y - 780 = 0

  2. Solve the quadratic equation using the quadratic formula y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

    Here, a=1a=1, b=4b=4, and c=−780c=-780.

    y=−4±42−4(1)(−780)2(1)y = \frac{-4 \pm \sqrt{4^2 - 4(1)(-780)}}{2(1)} y=−4±16+31202y = \frac{-4 \pm \sqrt{16 + 3120}}{2} y=−4±31362y = \frac{-4 \pm \sqrt{3136}}{2}

  3. Calculate the square root.

    3136=56\sqrt{3136} = 56

  4. Substitute the value back into the formula and find the two possible solutions for yy.

    y=−4±562y = \frac{-4 \pm 56}{2}

    For the first solution:

    y1=−4+562=522=26y_1 = \frac{-4 + 56}{2} = \frac{52}{2} = 26

    For the second solution:

    y2=−4−562=−602=−30y_2 = \frac{-4 - 56}{2} = \frac{-60}{2} = -30

  5. Choose the valid solution.

    Since age cannot be negative, we discard y=−30y=-30.
    Therefore, Victor's age is 26 years.

    Victor's age is:

    26 years old\boxed{26 \text{ years old}}
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