Don't understand quadratic word problems at all. Please help, with steps.
One number is 8 more than another number. Their product is 273. find the numbers.
The differance between a number and twice it's reciprocal is 17/10. What is the number?
Thanks,
Kelly
y = x + 8
x*y = 273
x*(x+8) = 273
x^2 + 8x - 273 = 0
(x-13)(x+21) = 0
One of those factors must evaluate to 0 for the answer to be 0. Therefore, x must equal either 13 or -21.
y must equal either 21 or -13.
So: x=13 and y=21
or x=-21 and y=-13
I'm sure you can figure out the second one.
For the first problem:
Let x = lesser number
let y = greater number
(x) (y) = 273
The problem states that y = x + 8.
substituting for y:
(x) (x+8)=273
x^2 + 8x=273
so
x^2 + 8x - 273 = 0
Solve for x.
Then, y = x + 8
To solve quadratic word problems, we need to follow a few steps. Let's start with the first problem and then move on to the second one.
1) One number is 8 more than another number. Their product is 273. Find the numbers.
Let's assume the first number is x and the second number is y.
Given that "One number is 8 more than another number," we can write an equation as:
x = y + 8
Also, we know that "Their product is 273," which can be written as:
x * y = 273
Now we have a system of equations:
x = y + 8 (equation 1)
x * y = 273 (equation 2)
To solve this system of equations, we can use the substitution method.
Step 1: Substitute equation 1 into equation 2:
(y + 8) * y = 273
Step 2: Simplify and rewrite the equation:
y^2 + 8y = 273
Step 3: Rearrange the equation:
y^2 + 8y - 273 = 0
Step 4: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this example, we'll use factoring:
Factor the quadratic equation:
(y - 13)(y + 21) = 0
Setting each factor equal to zero gives us two possibilities:
y - 13 = 0 or y + 21 = 0
Solve for y:
y = 13 or y = -21
Step 5: Substitute the values of y back into equation 1 to find the corresponding values of x:
For y = 13, x = 13 + 8 = 21
For y = -21, x = -21 + 8 = -13
So, the solution is (x, y) = (21, 13) or (-13, -21). These are the two numbers.
Now let's move on to the second problem:
2) The difference between a number and twice its reciprocal is 17/10. What is the number?
Let's assume the number is x.
According to the problem statement, "The difference between a number and twice its reciprocal is 17/10." This can be written as:
x - 2(1/x) = 17/10
To solve this equation, we need to simplify and rearrange it.
Step 1: Multiply the entire equation by 10x to eliminate the denominators:
10x(x) - 10x(2(1/x)) = (17/10)(10x)
Step 2: Simplify the equation:
10x^2 - 20 = 17x
Step 3: Rearrange the equation:
10x^2 - 17x - 20 = 0
Step 4: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this example, we'll use factoring:
Factor the quadratic equation:
(2x + 5)(5x - 4) = 0
Setting each factor equal to zero gives us two possibilities:
2x + 5 = 0 or 5x - 4 = 0
Solve for x:
2x = -5 or 5x = 4
x = -5/2 or x = 4/5
So, the solutions are x = -5/2 or x = 4/5. These are the possible numbers.
Keep in mind that the steps I provided are general guidelines for solving quadratic word problems. Depending on the specific problem, there may be variations in the steps required.