Math Problem Statement

Find a quadratic polynomial whose sum and product is the zeroes are -2√3 and -9.find it's zeroes

Solution

To find the quadratic polynomial whose sum and product of the zeroes are given, we can use the general form of a quadratic equation:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

For a quadratic equation with roots α\alpha and β\beta, we know that:

  1. Sum of the roots α+β=ba\alpha + \beta = -\frac{b}{a}
  2. Product of the roots αβ=ca\alpha \beta = \frac{c}{a}

Step 1: Use the given values

The sum of the roots is given as α+β=23\alpha + \beta = -2\sqrt{3}, and the product of the roots is αβ=9\alpha \beta = -9.

We can form the quadratic equation using these relationships:

f(x)=a(x2(α+β)x+αβ)f(x) = a(x^2 - (\alpha + \beta)x + \alpha \beta)

Substitute the values:

f(x)=a(x2(23)x+(9))f(x) = a\left(x^2 - (-2\sqrt{3})x + (-9)\right) f(x)=a(x2+23x9)f(x) = a\left(x^2 + 2\sqrt{3}x - 9\right)

To simplify, let a=1a = 1 for simplicity (this will give the quadratic polynomial without any scale factor). So, the polynomial becomes:

f(x)=x2+23x9f(x) = x^2 + 2\sqrt{3}x - 9

Step 2: Find the zeroes

To find the zeroes of the quadratic polynomial, we solve the equation:

x2+23x9=0x^2 + 2\sqrt{3}x - 9 = 0

We can use the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here, a=1a = 1, b=23b = 2\sqrt{3}, and c=9c = -9. Substituting these into the quadratic formula:

x=23±(23)24(1)(9)2(1)x = \frac{-2\sqrt{3} \pm \sqrt{(2\sqrt{3})^2 - 4(1)(-9)}}{2(1)} x=23±12+362x = \frac{-2\sqrt{3} \pm \sqrt{12 + 36}}{2} x=23±482x = \frac{-2\sqrt{3} \pm \sqrt{48}}{2} x=23±432x = \frac{-2\sqrt{3} \pm 4\sqrt{3}}{2}

Now, split into two cases:

  1. x=23+432=232=3x = \frac{-2\sqrt{3} + 4\sqrt{3}}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3}
  2. x=23432=632=33x = \frac{-2\sqrt{3} - 4\sqrt{3}}{2} = \frac{-6\sqrt{3}}{2} = -3\sqrt{3}

Final Answer:

The zeroes of the quadratic polynomial f(x)=x2+23x9f(x) = x^2 + 2\sqrt{3}x - 9 are:

x=3andx=33x = \sqrt{3} \quad \text{and} \quad x = -3\sqrt{3}

Would you like further details or assistance with another problem? Here are some related questions to explore:

  1. How does the discriminant of a quadratic equation help in determining the number and type of roots?
  2. Can the quadratic formula be applied when the coefficients are irrational numbers, like 232\sqrt{3}?
  3. What is the geometric interpretation of the zeroes of a quadratic polynomial?
  4. How can the roots of a quadratic be used to graph the parabola?
  5. What is the significance of the leading coefficient aa in a quadratic equation?

Tip: Always check that the sum and product of the roots match the coefficients of the quadratic equation for verification!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Roots of Quadratic Equations

Formulas

Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Vieta's formulas
Quadratic formula

Suitable Grade Level

Grades 9-11