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Solve Quadratic Equation 4m²=(m+4)(2-m) Using Quadratic Formula
Mathematics
Grade 9 (Junior High School)
Question Content
Solve the quadratic equation by formula: $4m^2=(m+4)(2-m)$
Correct Answer
$m = \\frac{-3 + \\sqrt{41}}{5}$ or $m = \\frac{-3 - \\sqrt{41}}{5}$
Detailed Solution Steps
1
Step 1: Expand and rearrange the equation to standard quadratic form $ax^2+bx+c=0$. First expand the right-hand side: $(m+4)(2-m)=2m - m^2 + 8 - 4m = -m^2 -2m +8$. Bring all terms to the left-hand side: $4m^2 + m^2 +2m -8 = 0$, which simplifies to $5m^2 +2m -8 = 0$.
2
Step 2: Identify the coefficients $a=5$, $b=2$, $c=-8$ from the standard form.
3
Step 3: Substitute into the quadratic formula $m = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}$. Calculate the discriminant first: $\\Delta = b^2-4ac = 2^2 - 4\\times5\\times(-8) = 4 + 160 = 164$.
4
Step 4: Simplify the square root of the discriminant: $\\sqrt{164}=\\sqrt{4\\times41}=2\\sqrt{41}$.
5
Step 5: Substitute back into the formula: $m = \\frac{-2 \\pm 2\\sqrt{41}}{2\\times5} = \\frac{-2 \\pm 2\\sqrt{41}}{10}$. Factor out 2 from the numerator and cancel with the denominator: $m = \\frac{-1 \\pm \\sqrt{41}}{5}$.
Knowledge Points Involved
1
Standard Form of Quadratic Equations
A quadratic equation is written in the standard form $ax^2+bx+c=0$ where $a$, $b$, $c$ are constants and $a\\neq0$. This form is required to use the quadratic formula, as it clearly defines the coefficients needed for calculation.
2
Quadratic Formula
The quadratic formula $x = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}$ is used to find the roots of any quadratic equation in standard form. The term $b^2-4ac$ is called the discriminant, which determines the nature of the roots (real, repeated, or complex).
3
Expanding Polynomial Products
To expand products like $(m+4)(2-m)$, use the distributive property (FOIL method for binomials): multiply the First terms, Outer terms, Inner terms, and Last terms, then combine like terms to simplify the expression.
4
Simplifying Radicals
When simplifying square roots, factor out perfect squares from the radicand. For example, $\\sqrt{164}=\\sqrt{4\\times41}=2\\sqrt{41}$, since $\\sqrt{4}=2$ is a perfect square root.
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