Question Find your age if $x^2 - 28x = 60$ , where $x$ represents your age.

Studdy Solution

STEP 1

Assumptions
1. Let the age be represented by the variable $x$ .
2. The equation that represents the problem is $x^2 - 28x = 60$ .

STEP 2

First, we need to set up the equation according to the given problem.
$x^2 - 28x = 60$

STEP 3

To solve for $x$ , we should move all terms to one side of the equation to set it equal to zero. This will allow us to factor or use the quadratic formula.
$x^2 - 28x - 60 = 0$

STEP 4

Now, we need to factor the quadratic equation if possible. We look for two numbers that multiply to $-60$ and add up to $-28$ .

STEP 5

We find that the numbers $-30$ and $+2$ satisfy the conditions because $-30 \times 2 = -60$ and $-30 + 2 = -28$ .

STEP 6

We can now factor the quadratic equation using these two numbers.
$x^2 - 30x + 2x - 60 = 0$

STEP 7

Next, we group the terms to factor by grouping.
$(x^2 - 30x) + (2x - 60) = 0$

STEP 8

Factor out the common factors in each group.
$x(x - 30) + 2(x - 30) = 0$

STEP 9

Now, we see that $(x - 30)$ is a common factor.
$(x + 2)(x - 30) = 0$

STEP 10

According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.
$x + 2 = 0 \quad \text{or} \quad x - 30 = 0$

STEP 11

Solve each equation for $x$ .
For $x + 2 = 0$ : $x = -2$
For $x - 30 = 0$ : $x = 30$

STEP 12

Since age cannot be negative, we discard the negative solution.
The age is $x = 30$ .
Karan's age is 30 years old.

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