Examkin

Question

Difficulty: MediumQuadratic Equations

In the quadratic equation 2x2−16x+c=02x^2 - 16x + c = 0, cc is a constant. If one of the solutions to the equation is 4−34 - \sqrt{3}, what is the value of cc?

Answer: 26

Answer

26
Since the coefficients of the quadratic equation are real, any irrational roots must occur in conjugate pairs. Therefore, if one solution is 4−34 - \sqrt{3}, the other solution must be 4+34 + \sqrt{3}. According to Vieta's formulas, the product of the roots r1r_1 and r2r_2 for a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is ca\frac{c}{a}. In this equation, a=2a = 2, so the product of the roots is c2\frac{c}{2}. Multiplying the two solutions gives (4−3)(4+3)=16−3=13(4 - \sqrt{3})(4 + \sqrt{3}) = 16 - 3 = 13. Setting c2=13\frac{c}{2} = 13 and solving for cc yields c=26c = 26.

Step-by-Step Solution

1
Determine the second root of the quadratic equation.
The second root is 4+34 + \sqrt{3}.
Since the quadratic equation has real coefficients, the irrational roots must be conjugate pairs.
2
Express the product of the roots using Vieta's formulas.
The product of the roots is c2\frac{c}{2}.
For any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the product of the roots is equal to ca\frac{c}{a}. Here, a=2a = 2.
3
Calculate the product of the two roots.
(4−3)(4+3)=16−3=13(4 - \sqrt{3})(4 + \sqrt{3}) = 16 - 3 = 13.
Using the difference of squares formula, (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.
4
Equate the product of the roots to the expression from step 2 and solve for cc.
c2=13⇒c=26\frac{c}{2} = 13 \Rightarrow c = 26.
By substituting the calculated product into the formula for the product of the roots, we find the value of the constant.

Key Concept

Using Vieta's formulas and the conjugate root theorem to solve for coefficients of a quadratic equation.

Alternative Method

Alternatively, substitute the given solution x=4−3x = 4 - \sqrt{3} directly into the equation 2x2−16x+c=02x^2 - 16x + c = 0 and solve for cc. First, calculate x2=(4−3)2=16−83+3=19−83x^2 = (4 - \sqrt{3})^2 = 16 - 8\sqrt{3} + 3 = 19 - 8\sqrt{3}. Then substitute this into the equation: 2(19−83)−16(4−3)+c=02(19 - 8\sqrt{3}) - 16(4 - \sqrt{3}) + c = 0. Simplifying this yields 38−163−64+163+c=0⇒−26+c=0⇒c=2638 - 16\sqrt{3} - 64 + 16\sqrt{3} + c = 0 \Rightarrow -26 + c = 0 \Rightarrow c = 26.
Estimated Time:1m 30s
Rate this question

Topics

Quadratic Equations

More questions from this topic