Create a quadratic polynomial whose sum and product of zeros are 1/4 and -1 respectively.

Mathematics
Create a quadratic polynomial whose sum and product of zeros are 1/4 and -1 respectively.

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Answer

4x2−x−14x^{2} - x - 1

Step 1: Recall the sum and product of zeros for the quadratic ax2+bx+c=0ax^2 + bx + c = 0.

Sum of zeros: s=−bas = -\frac{b}{a}

Product of zeros: p=cap = \frac{c}{a}

Given: s=14s = \frac{1}{4}, p=−14p = -\frac{1}{4}

s=−ba=14s = -\frac{b}{a} = \frac{1}{4} p=ca=−14p = \frac{c}{a} = -\frac{1}{4}

Step 2: Choose a=4a = 4 to obtain integer coefficients.

Substitute a=4a = 4 into sum formula:

−b4=14-\frac{b}{4} = \frac{1}{4}

Multiply both sides by -4:

b=−1b = -1

Step 3: Substitute a=4a = 4 into product formula:

c4=−14\frac{c}{4} = -\frac{1}{4}

Multiply both sides by 4:

c=−1c = -1

Step 4: Form the quadratic polynomial.

4x2+(−1)x+(−1)=4x2−x−14x^2 + (-1)x + (-1) = 4x^2 - x - 1

Step 5: Verify sum and product.

Sum: −ba=−−14=14-\frac{b}{a} = -\frac{-1}{4} = \frac{1}{4}

Product: ca=−14=−14\frac{c}{a} = \frac{-1}{4} = -\frac{1}{4}

4x^{2 - x - 1}

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Quick Answer

Recall the sum and product of zeros for the quadratic ax^2 + bx + c = 0 .

Create a quadratic polynomial whose sum and product of zeros are 1/4 and -1 respectively.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Recall the sum and product of zeros for the quadratic ax^2 + bx + c = 0 . Sum of zeros: s = -(b)/(a) Product of zeros: p = (c)/(a) Given: s = (1)/(4) , p = -(1)/(4) s = -(b)/(a) = (1)/(4) p = (c)/(a) = -(1)/(4) Step 2: Choose a = 4 to obtain integer coefficients. Substitute a = 4 into sum formula: -(b)/(4) = (1)/(4) Multiply both sides by -4: b = -1 Step 3: Substitute a = 4 into product formula: (c)/(4) = -(1)/(4) Multiply both sides by 4: c = -1 Step 4: Form the quadratic polynomial. 4x^2 + (-1)x + (-1) = 4x^2 - x - 1 Step 5: Verify sum and product. Sum: -(b)/(a) = -(-1)/(4) = (1)/(4) Product: (c)/(a) = (-1)/(4) = -(1)/(4) 4x^2 - x - 1