Solve: 4a^2+19a+21 = 0
Answer
a = -1.75 or a = -3
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Step-by-Step Breakdown
- Expand and simplify the left-hand side: 4a^2+19a+21 = 4a^2 + 19a + 21
- Expand and simplify the right-hand side: 0 = 0
- Move every term to one side so the equation reads "...= 0": 4a^2 + 19a + 21 = 0
- This is a quadratic equation of the form aa^2+ba+c=0, with a=4, b=19, c=21.
- Compute the discriminant: D = b^2-4ac = (19)^2 - 4(4)(21) = 25
- D > 0, so there are two real solutions: a = (-b ± √D) / (2a) = (-1.75, -3)
Theory & Definitions
A quadratic equation aa^2+ba+c=0 (a≠0) is solved with the quadratic formula: a = (-b ± √(b^2-4ac)) / (2a). The discriminant D=b^2-4ac tells you how many real solutions exist: two when D>0, one repeated solution when D=0, and two complex solutions when D<0.
Frequently Asked Questions
What is the solution to 4a^2+19a+21 = 0?
a = -1.75 or a = -3.