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Magazine Chapter 9: Problem 113
Solve by using the Quadratic Formula. $$4 m^{2}+m-3=0$$
Short Answer
Expert verified
The solutions are \(m = \frac{3}{4}\) and \(m = -1\).
Step by step solution
01
Identify coefficients
The quadratic equation given is in the form of \[4m^{2} + m - 3 = 0\]. In a quadratic equation of the standard form \[ax^2 + bx + c = 0\], identify the coefficients: \[a = 4, \, b = 1, \, c = -3\].
02
Recall the Quadratic Formula
The Quadratic Formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula will be used to find the values of \(m\).
03
Substitute coefficients into Quadratic Formula
Substitute \(a = 4\), \(b = 1\), and \(c = -3\) into the Quadratic Formula: \[m = \frac{-(1) \pm \sqrt{(1)^2 - 4(4)(-3)}}{2(4)}\].
04
Simplify inside the square root
Calculate the expression under the square root: \(b^2 - 4ac\): \[1 - 4 \cdot 4 \cdot (-3) = 1 + 48 = 49\]. So, \[m = \frac{-1 \pm \sqrt{49}}{8}\].
05
Solve for both values of \(m\)
Solve for \(m\) by evaluating the \(\pm\) cases: \[m = \frac{-1 \pm 7}{8}\].
06
Evaluate the positive root
Evaluate \(m = \frac{-1 + 7}{8}\): \[m = \frac{6}{8} = \frac{3}{4}\].
07
Evaluate the negative root
Evaluate \(m = \frac{-1 - 7}{8}\): \[m = \frac{-8}{8} = -1\].
08
Write the final solution
The solutions for the quadratic equation \(4m^2 + m - 3 = 0\) are \(m = \frac{3}{4}\) and \(m = -1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quadratic equations
A quadratic equation is a type of polynomial equation of the second degree. This means its highest exponent of the variable is 2. The standard form of a quadratic equation is given by \[ ax^2 + bx + c = 0 \], where
- \(a\), \(b\), and \(c\) are coefficients
- \(a eq 0\)
solving quadratic equations
There are several methods to solve quadratic equations, including factoring, completing the square, and using the Quadratic Formula. The Quadratic Formula is a powerful tool since it can solve any quadratic equation, regardless of whether it can be factored. The formula is given by: \[ x = \frac{-b \,\pm\, \sqrt{b^2 - 4ac}}{2a} \]. Here, \(b^2 - 4ac\) is known as the discriminant. The value of the discriminant determines the nature of the roots:
- If \(b^2 - 4ac > 0\), there are two distinct real solutions
- If \(b^2 - 4ac = 0\), there is one real solution (repeated root)
- If \(b^2 - 4ac < 0\), there are two complex solutions
coefficients in algebra
In algebra, coefficients are the numerical factors multiplying the variables in an equation. In a quadratic equation of form \[ ax^2 + bx + c = 0 \], the coefficients are:
- \(a\) is the coefficient of \(x^2\) term
- \(b\) is the coefficient of \(x\) term
- \(c\) is the constant term
