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Solve the quadratic equation 4m²=(m+4)(2-m) using algebraic formulas
Mathematics
Grade 8 of Junior High School
Question Content
Solve the given equation/identity by using formula: $4m^2=(m+4)(2-m)$
Correct Answer
$m = \\frac{-2 + 2\\sqrt{5}}{3}$ or $m = \\frac{-2 - 2\\sqrt{5}}{3}$
Detailed Solution Steps
1
Step 1: Expand the right-hand side of the equation using the distributive property (FOIL method): $(m+4)(2-m) = m*2 - m*m + 4*2 - 4*m = 2m - m^2 + 8 - 4m = -m^2 -2m +8$
2
Step 2: Rearrange the original equation to standard quadratic form $ax^2+bx+c=0$: Move all terms to the left-hand side: $4m^2 + m^2 +2m -8 = 0$, which simplifies to $5m^2 +2m -8 = 0$
3
Step 3: Use the quadratic formula $m = \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$. Here, $a=5$, $b=2$, $c=-8$
4
Step 4: Calculate the discriminant first: $\\Delta = b^2-4ac = 2^2 - 4*5*(-8) = 4 + 160 = 164 = 4*41$
5
Step 5: Substitute into the quadratic formula: $m = \\frac{-2\\pm\\sqrt{164}}{2*5} = \\frac{-2\\pm2\\sqrt{41}}{10} = \\frac{-1\\pm\\sqrt{41}}{5}$
Knowledge Points Involved
1
FOIL Method for Binomial Multiplication
A technique to multiply two binomials: multiply the First terms, Outer terms, Inner terms, and Last terms, then sum the results. Used to expand products of linear expressions, which is essential for rearranging polynomial equations.
2
Standard Quadratic Form
The standard form of a quadratic equation is $ax^2+bx+c=0$ where $a$, $b$, $c$ are constants and $a\\neq0$. This form is required to apply the quadratic formula to find the roots of the equation.
3
Quadratic Formula
A formula $x = \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ that gives the solutions (roots) of any quadratic equation in standard form. The discriminant $b^2-4ac$ determines the nature of the roots (real, repeated, or complex).
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