Q. \({x}^{2}-x-30\)

Q: What is the vertex of y=x^2-x-30 and its minimum value?

Vertex at x=-\frac{b}{2a}=\frac{1}{2}. Minimum y=\left(\frac{1}{2}\right)^2-\frac{1}{2}-30=-\frac{121}{4}.

Q: What is the discriminant of x^2-x-30?

Discriminant b^2-4ac=(-1)^2-4(1)(-30)=121.

Q: Determine the y-intercept of x^2-x-30.

Set x=0: y=-30, so intercept is (0,-30).

Answer

We factor the quadratic \(x^2-x-30\) by finding two numbers that multiply to \(-30\) and add to \(-1\). Those numbers are \(-6\) and \(5\).

\[
x^2-x-30=(x-6)(x+5)
\]

So the zeros are determined by \((x-6)=0\) or \((x+5)=0\), giving \(x=6\) or \(x=-5\).

Final result: \(x=6\) or \(x=-5\).

Detailed Explanation

We want to factor the expression \(x^2 – x – 30\).

Step 1: Identify the coefficients.

The expression is in the form \(ax^2 + bx + c\).

\(a = 1\), \(b = -1\), \(c = -30\).

Step 2: Find two numbers that multiply to \(ac\) and add to \(b\).

Because \(a = 1\), we need two integers \(m\) and \(n\) such that:

\(m \cdot n = -30\)
\(m + n = -1\)

Step 3: List factor pairs of \(-30\).

Factor pairs of \(-30\) are:

\(-1\) and \(30\) (sum \(= 29\))
\(1\) and \(-30\) (sum \(= -29\))
\(-2\) and \(15\) (sum \(= 13\))
\(2\) and \(-15\) (sum \(= -13\))
\(-3\) and \(10\) (sum \(= 7\))
\(3\) and \(-10\) (sum \(= -7\))
\(-5\) and \(6\) (sum \(= 1\))
\(5\) and \(-6\) (sum \(= -1\))

The pair that works is \(5\) and \(-6\), because:

\(5 \cdot (-6) = -30\)
\(5 + (-6) = -1\)

Step 4: Write the factored form.

Since \(5\) and \(-6\) are our numbers, we split the middle term \(-x\) as \(5x – 6x\):

\[
x^2 – x – 30 = x^2 + 5x – 6x – 30
\]

Now factor by grouping:

\[
x^2 + 5x – 6x – 30 = (x^2 + 5x) + (-6x – 30)
\]
\[
= x(x + 5) – 6(x + 5)
\]

Factor out \((x + 5)\):

\[
= (x + 5)(x – 6)
\]

Final Answer:

\[
x^2 – x – 30 = (x + 5)(x – 6)
\]

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Algebra FAQ

What are the factors of \(x^2-x-30\)?

Factor as \((x-6)(x+5)\).

Solve \(x^2-x-30=0\).

\((x-6)(x+5)=0\) so \(x=6\) or \(x=-5\).

What are the zeros of \(x^2-x-30\)?

The zeros are \(x=6\) and \(x=-5\).

Use the quadratic formula for \(x^2-x-30=0\).

\(x=\frac{1\pm\sqrt{1-4(1)(-30)}}{2}=\frac{1\pm\sqrt{121}}{2}=\frac{1\pm 11}{2}\), giving \(x=6,-5\).

What is the vertex of \(y=x^2-x-30\) and its minimum value?

Vertex at \(x=-\frac{b}{2a}=\frac{1}{2}\). Minimum \(y=\left(\frac{1}{2}\right)^2-\frac{1}{2}-30=-\frac{121}{4}\).

What is the discriminant of \(x^2-x-30\)?

Discriminant \(b^2-4ac=(-1)^2-4(1)(-30)=121\).

Determine the \(y\)-intercept of \(x^2-x-30\).

Set \(x=0\): \(y=-30\), so intercept is \((0,-30)\).

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