Q. Determine the x-intercepts of the following equation. \( (2x+4)(4x-12)=y \)

Q: How do I find the x-intercepts of (2x+4)(4x-12)=y ?

Set y=0: (2x+4)(4x-12)=0. Solve each factor: 2x+4=0 \Rightarrow x=-2; 4x-12=0 \Rightarrow x=3. X-intercepts: (-2,0) and (3,0).

Q: Should I simplify factors before solving?

You can, but it’s optional. For 2x+4=0 divide by 2 to get x=-2. For 4x-12=0 divide by 4 to get x=3. Simplifying makes arithmetic easier but doesn’t change solutions.

Q: What is the y-intercept of this function?

Set x=0: y=(2\cdot0+4)(4\cdot0-12)=4\cdot(-12)=-48. Y-intercept is (0,-48).

Q: How would I graph this quickly?

Plot x-intercepts (-2,0), (3,0) and y-intercept (0,-48); note the parabola opens upward (leading term 8x^2 positive). Sketch a smooth curve through these points..

Answer

Set y = 0: \(0=(2x+4)(4x-12)\).

Solve factors: \(2x+4=0\) so \(x=-2\). \(4x-12=0\) so \(x=3\).

x-intercepts: \((-2,0)\) and \((3,0)\).

Detailed Explanation

Definition: An x-intercept is a point where the graph crosses the x-axis, so the y-coordinate is 0. Set y equal to 0 in the given equation.
\[ (2x+4)(4x-12)=0 \]
Apply the Zero-Product Property: If a product of two factors equals 0, then at least one of the factors must be 0. Set each factor equal to 0 separately.
\[ 2x+4=0 \quad \text{or} \quad 4x-12=0 \]
Solve the first equation 2x + 4 = 0.
Subtract 4 from both sides:

\[ 2x = -4 \]

Divide both sides by 2:

\[ x = -2 \]
Solve the second equation 4x − 12 = 0.
Add 12 to both sides:

\[ 4x = 12 \]

Divide both sides by 4:

\[ x = 3 \]
Conclusion: The x-intercepts are the points with these x-values and y = 0.
\[ (-2,\,0) \quad \text{and} \quad (3,\,0) \]

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Algebra FAQs

What does x-intercept mean?

The x-intercepts are points where the graph crosses the x-axis, so \(y=0\). Solve the equation with \(y=0\) to find the x-values; each gives a point \((x,0)\).

How do I find the x-intercepts of \( (2x+4)(4x-12)=y \)?

Set \(y=0\): \((2x+4)(4x-12)=0\). Solve each factor: \(2x+4=0 \Rightarrow x=-2\); \(4x-12=0 \Rightarrow x=3\). X-intercepts: \((-2,0)\) and \((3,0)\).

Why is it valid to set each factor equal to zero?

If a product equals zero, at least one factor must be zero. So solving each linear factor yields all solutions for \(y = 0\). This follows from the zero-product property.

Should I simplify factors before solving?

You can, but it’s optional. For \(2x+4=0\) divide by \(2\) to get \(x=-2\). For \(4x-12=0\) divide by \(4\) to get \(x=3\). Simplifying makes arithmetic easier but doesn’t change solutions.

Do multiplicities matter for x-intercepts here?

Yes: a root’s multiplicity affects whether the graph crosses or just touches the axis. Both factors are linear (multiplicity 1), so the graph crosses the x-axis at \((-2,0)\) and \((3,0)\).

How do I check my answers?

What is the \(y\)-intercept of this function?

Set \(x=0\): \(y=(2\cdot0+4)(4\cdot0-12)=4\cdot(-12)=-48\). Y-intercept is \((0,-48)\).

How would I graph this quickly?

Plot x-intercepts \((-2,0)\), \((3,0)\) and y-intercept \((0,-48)\); note the parabola opens upward (leading term \(8x^2\) positive). Sketch a smooth curve through these points..

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