Math Problem Statement

Write a quadratic function with zeros 0 and – 5. Write your answer using the variable x and in standard form with a leading coefficient of 1.

Solution

To write a quadratic function with zeros 00 and 5-5, we use the fact that if aa and bb are zeros of a quadratic function, the function can be written as:

f(x)=a(xr1)(xr2)f(x) = a(x - r_1)(x - r_2)

Where r1r_1 and r2r_2 are the zeros. Here, r1=0r_1 = 0 and r2=5r_2 = -5.

Substitute the zeros into the equation:

f(x)=a(x0)(x(5))f(x) = a(x - 0)(x - (-5))

f(x)=a(x)(x+5)f(x) = a(x)(x + 5)

Since the leading coefficient a=1a = 1, the equation becomes:

f(x)=x(x+5)f(x) = x(x + 5)

Expand to write it in standard form:

f(x)=x2+5xf(x) = x^2 + 5x

Final Answer:

f(x)=x2+5xf(x) = x^2 + 5x

Do you want me to explain why this works or how to adjust for different coefficients?
Here are 5 related questions to expand your understanding:

  1. What would the quadratic function look like if the zeros were 22 and 3-3?
  2. How do you determine the vertex of a quadratic function from its standard form?
  3. Can you find the axis of symmetry for this quadratic function?
  4. How does the graph of this quadratic function relate to its factored form?
  5. How do you find the y-intercept of a quadratic function in standard form?

Tip: To check if a quadratic function has the correct zeros, substitute each zero back into the function and ensure it equals zero!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Factored Form of Polynomials

Formulas

Standard form of a quadratic equation: ax^2 + bx + c = 0
Factored form of a quadratic equation: a(x - r1)(x - r2)

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 8-10