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Difficulty: MediumQuadratic Equations and the Quadratic Formula

The quadratic equation 0.4x2−2x+c=00.4x^2 - 2x + c = 0 has two real solutions. If the difference between these two solutions is exactly 33, what is the value of cc?

Answer: 1.6

Answer

1.6
By using the relation between the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, we find that the difference of the roots is given by ∣x1−x2∣=b2−4ac∣a∣|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}. Substituting a=0.4a = 0.4, b=−2b = -2, and the difference of 33 yields 3=4−1.6c0.43 = \frac{\sqrt{4 - 1.6c}}{0.4}. Multiplying by 0.40.4 and squaring both sides gives 1.44=4−1.6c1.44 = 4 - 1.6c. Solving this linear equation gives c=1.6c = 1.6. We can verify this result by substituting c=1.6c = 1.6 back into the original equation: 0.4x2−2x+1.6=00.4x^2 - 2x + 1.6 = 0 simplifies to x2−5x+4=0x^2 - 5x + 4 = 0, which factors as (x−1)(x−4)=0(x-1)(x-4) = 0. The roots are 11 and 44, and their difference is 4−1=34 - 1 = 3.

Step-by-Step Solution

1
Identify the coefficients of the quadratic equation.
a=0.4a = 0.4, b=−2b = -2, and the constant term is cc.
To apply formulas relating the roots to the coefficients of the equation.
2
Apply the formula for the difference of the roots.
∣x1−x2∣=b2−4ac∣a∣|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}
The problem states the difference between the two solutions is 33.
3
Substitute the known values into the formula and solve.
3=(−2)2−4(0.4)c0.4⇒1.2=4−1.6c3 = \frac{\sqrt{(-2)^2 - 4(0.4)c}}{0.4} \Rightarrow 1.2 = \sqrt{4 - 1.6c}
To isolate the square root expression containing the unknown variable.
4
Square both sides and solve the linear equation for cc.
1.44=4−1.6c⇒1.6c=2.56⇒c=1.61.44 = 4 - 1.6c \Rightarrow 1.6c = 2.56 \Rightarrow c = 1.6
To eliminate the square root and find the value of cc.

Key Concept

Quadratic Equations and the Quadratic Formula
Estimated Time:1m 30s
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Topics

Quadratic Equations and the Quadratic Formula

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