Solved on Dec 11, 2023

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

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

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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Find your age if x2−28x=60x^2 - 28x = 60x2−28x=60, where xxx represents your age.

STEP 1

Assumptions1. Let the age be represented by the variable xxx.2. The equation that represents the problem is x2−28x=60x^2 - 28x = 60x2−28x=60.

STEP 2

First, we need to set up the equation according to the given problem.x2−28x=60x^2 - 28x = 60x2−28x=60

STEP 3

To solve for xxx, 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.x2−28x−60=0x^2 - 28x - 60 = 0x2−28x−60=0

STEP 4

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

STEP 5

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

STEP 6

We can now factor the quadratic equation using these two numbers.x2−30x+2x−60=0x^2 - 30x + 2x - 60 = 0x2−30x+2x−60=0

STEP 7

Next, we group the terms to factor by grouping.(x2−30x)+(2x−60)=0(x^2 - 30x) + (2x - 60) = 0(x2−30x)+(2x−60)=0

STEP 8

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

STEP 9

Now, we see that (x−30)(x - 30)(x−30) is a common factor.(x+2)(x−30)=0(x + 2)(x - 30) = 0(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=0orx−30=0x + 2 = 0 \quad \text{or} \quad x - 30 = 0x+2=0orx−30=0

STEP 11

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

STEP 12

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

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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

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

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.
$x^2 - 28x - 60 = 0$

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

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

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

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

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

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

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

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

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