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Difficulty: EasyQuadratic Equations and the Quadratic Formula

What is the sum of the two solutions to the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0?

Answer: 3.5

Answer

The sum of the solutions is 3.53.5.
The sum of the solutions of the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0 is 3.53.5. According to Vieta's formulas, the sum of the roots of a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0 is given by −ba-\frac{b}{a}. Here, a=2a = 2 and b=−7b = -7, so the sum is −−72=3.5-\frac{-7}{2} = 3.5. Alternatively, solving the equation using the quadratic formula yields the roots 44 and −0.5-0.5, which sum to 3.53.5.

Step-by-Step Solution

1
Identify the coefficients from the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0.
a=2a = 2, b=−7b = -7, and c=−4c = -4.
A quadratic equation in standard form is written as ax2+bx+c=0ax^2 + bx + c = 0.
2
Apply the sum of roots formula −ba-\frac{b}{a}.
Sum =−−72=3.5= -\frac{-7}{2} = 3.5.
By Vieta's formulas, the sum of the roots of ax2+bx+c=0ax^2 + bx + c = 0 is −ba-\frac{b}{a}.

Key Concept

Sum of roots of a quadratic equation using Vieta's formulas

Alternative Method

Solve the quadratic equation by factoring or using the quadratic formula. Factoring 2x2−7x−4=02x^2 - 7x - 4 = 0 yields (2x+1)(x−4)=0(2x + 1)(x - 4) = 0, which gives solutions x=−0.5x = -0.5 and x=4x = 4. Adding these solutions together gives −0.5+4=3.5-0.5 + 4 = 3.5.
Estimated Time:1m 0s
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Topics

Quadratic Equations and the Quadratic Formula

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