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Magazine Chapter 4: Problem 45
Factor. \(6 x^{2}+7 x-10\)
Short Answer
Expert verified
The quadratic \(6x^2 + 7x - 10\) factors to \((6x - 5)(x + 2)\).
Step by step solution
01
Identify the form of the quadratic
The quadratic expression given is \(6x^2 + 7x - 10\). This expression is in the standard quadratic form \(ax^2 + bx + c\) where \(a = 6\), \(b = 7\), and \(c = -10\).
02
Multiply the leading coefficient and constant term
Compute the product of the leading coefficient \(a\) and the constant term \(c\): \(6 \times -10 = -60\). We need to find two numbers that multiply to \(-60\).
03
Find two numbers that multiply to -60 and add to 7
We need to find two numbers that multiply to \(-60\) and sum up to \(7\). These numbers are \(12\) and \(-5\) because \(12 \times (-5) = -60\) and \(12 + (-5) = 7\).
04
Rewrite the middle term using the two numbers
Rewrite the original expression by splitting the middle term using \(12\) and \(-5\): \(6x^2 + 12x - 5x - 10\).
05
Group and factor each pair
Group the first two terms and the last two terms: \((6x^2 + 12x) + (-5x - 10)\). Factor each group: \(6x(x + 2) - 5(x + 2)\).
06
Factor out the common binomial
Notice the common binomial \((x + 2)\) in both terms. Factor it out: \( (6x - 5)(x + 2) \). This is the factored form of the quadratic expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Quadratics
Factoring quadratics is like solving a puzzle. The goal is to rewrite a quadratic expression as a product of two binomials. For example, given a quadratic like \(6x^2 + 7x - 10\), the task is to express it in a simpler form. Factoring helps in solving quadratic equations and finding the roots. The key is to break down complex expressions into simpler ones.
To factor quadratics:
To factor quadratics:
- First, identify the quadratic expression as \(ax^2 + bx + c\).
- Then, find numbers that multiply and add up to specific values derived from the expression's coefficients.
- Rewrite the expression, grouping terms to facilitate factoring.
- Finally, factor the expression completely into a product of binomials.
Standard Quadratic Form
The standard quadratic form is a universal way to express quadratic equations. It's written as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Each part of this form has a role:
Recognizing the standard form is crucial because it tells you how to approach factoring. In our example \(6x^2 + 7x - 10\), identifying \(a = 6\), \(b = 7\), and \(c = -10\) sets the stage for factoring by grouping and finding roots.
- \(a\) is the leading coefficient, affecting the parabola's width and direction.
- \(b\) influences the parabola's axis of symmetry.
- \(c\) is the constant term, determining the parabola's vertical position.
Recognizing the standard form is crucial because it tells you how to approach factoring. In our example \(6x^2 + 7x - 10\), identifying \(a = 6\), \(b = 7\), and \(c = -10\) sets the stage for factoring by grouping and finding roots.
Leading Coefficient
The leading coefficient in a quadratic expression is the number in front of the \(x^2\) term. In \(6x^2 + 7x - 10\), it's \(6\). This coefficient plays a significant role:
- If it's positive, the parabola opens upwards; if negative, it opens downwards.
- The magnitude determines how "steep" or "wide" the parabola is.
Factoring by Grouping
Factoring by grouping involves rearranging and grouping terms to simplify a quadratic expression. It's effective when dealing with quadratics that have a leading coefficient other than one, like \(6x^2 + 7x - 10\). Follow these steps:
This method simplifies complex quadratics into an understandable form.
- Find two numbers that multiply to \(a\times c\) and add to \(b\).
- Use them to split the middle term, e.g., split \(7x\) into \(12x - 5x\).
- Rewrite and group terms: \((6x^2 + 12x) + (-5x - 10)\).
- Factor each group individually: \(6x(x + 2) - 5(x + 2)\).
- Factor out the common binomial factor, \((x + 2)\), to get the final factored expression: \((6x - 5)(x + 2)\).
This method simplifies complex quadratics into an understandable form.
