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Factor the quadratic expression 10x² -17x +6 | Math Problem Solution
Mathematics
Grade 9 (Junior High School)
Question Content
Factor the quadratic expression: $10x^2 - 17x + 6$, and choose the correct factored form from the options: $(2x - 6)(5x - 1)$, $(2x - 3)(5x - 2)$, $(2x - 1)(5x - 6)$, $(10x - 6)(x - 1)$
Correct Answer
$(2x - 3)(5x - 2)$
Detailed Solution Steps
1
Step 1: Use the AC method for factoring quadratic trinomials of the form $ax^2+bx+c$. Here, $a=10$, $b=-17$, $c=6$. Calculate $ac=10*6=60$.
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Step 2: Find two integers that multiply to $60$ and add up to $b=-17$. The integers are $-12$ and $-5$, since $(-12)*(-5)=60$ and $(-12)+(-5)=-17$.
3
Step 3: Rewrite the middle term using these two integers: $10x^2 -12x -5x +6$.
4
Step 4: Factor by grouping: Group the first two and last two terms: $(10x^2 -12x)+(-5x +6)$. Factor out the greatest common factor (GCF) from each group: $2x(5x -6) -1(5x -6)$.
5
Step 5: Factor out the common binomial factor $(5x -6)$ to get $(2x -1)(5x -6)$? No, correct the grouping check: Alternatively, test each option by expanding using the FOIL method. Expand $(2x - 3)(5x - 2)$: First terms $2x*5x=10x^2$, Outer terms $2x*(-2)=-4x$, Inner terms $-3*5x=-15x$, Last terms $-3*(-2)=6$. Combine like terms: $10x^2 -4x -15x +6=10x^2 -17x +6$, which matches the original expression.
6
Step 6: Verify other options are incorrect: Expanding $(2x -6)(5x -1)$ gives $10x^2-2x-30x+6=10x^2-32x+6$; $(2x -1)(5x -6)$ gives $10x^2-12x-5x+6=10x^2-17x+6$? Correction: Wait, this also matches? No, wait $-12x-5x=-17x$, yes. Wait no, original calculation: $10x^2-12x-5x+6=10x^2-17x+6$. But wait, $(2x-3)(5x-2)$ also gives $10x^2-4x-15x+6=10x^2-17x+6$. Wait, no, $-4x-15x=-19x$, mistake! Oh right, $2x*(-2)=-4x$, $-3*5x=-15x$, sum is $-19x$, which is wrong. The correct grouping is $10x^2-5x-12x+6$, group as $(10x^2-5x)+(-12x+6)=5x(2x-1)-6(2x-1)=(5x-6)(2x-1)$. Wait, but expanding $(2x-1)(5x-6)$ gives $10x^2-12x-5x+6=10x^2-17x+6$, which is correct. Wait, the earlier FOIL for $(2x-3)(5x-2)$ was wrong, it gives $10x^2-4x-15x+6=10x^2-19x+6$, which is incorrect. So the correct answer is $(2x-1)(5x-6)$? Wait no, wait the original expression is $10x^2-17x+6$. Let's recalculate the AC method: $ac=60$, find two numbers that multiply to 60 and add to -17: -12 and -5, yes. So rewrite as $10x^2-12x-5x+6$, group as $(10x^2-12x)+(-5x+6)=2x(5x-6)-1(5x-6)=(2x-1)(5x-6)$, which expands to $10x^2-12x-5x+6=10x^2-17x+6$, correct. The option $(2x-3)(5x-2)$ expands to $10x^2-4x-15x+6=10x^2-19x+6$, which is wrong. So the correct answer is $(2x-1)(5x-6)$.
Knowledge Points Involved
1
Factoring Quadratic Trinomials (AC Method)
A method for factoring trinomials of the form $ax^2+bx+c$ where $a\\neq1$. It involves finding two factors of the product $ac$ that add up to $b$, then rewriting the middle term using these factors and factoring by grouping. Used when the quadratic cannot be factored using simple integer pairs for the first and last terms directly.
2
FOIL Method for Expanding Binomials
A mnemonic for multiplying two binomials: First (multiply the first terms of each binomial), Outer (multiply the outer terms), Inner (multiply the inner terms), Last (multiply the last terms). It is used to verify factored forms by expanding them back to the original quadratic expression.
3
Factoring by Grouping
A technique where a polynomial with four terms is split into two pairs, the greatest common factor (GCF) is factored out of each pair, and then a common binomial factor is factored out. It is often used after rewriting the middle term in the AC method for factoring quadratics.
4
Greatest Common Factor (GCF)
The largest integer or monomial that divides all terms of a polynomial without a remainder. It is used to simplify polynomial expressions and is a key step in factoring by grouping.
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