Q. \(x^2 + 14x + 45\)

Q: What are the roots of x^{2}+14x+45=0 ?

From (x+5)(x+9)=0, x=-5 or x=-9.

Q: Solve using the quadratic formula for x^{2}+14x+45=0 ?

a=1,b=14,c=45. x=\frac{-14\pm\sqrt{196-180}}{2}=\frac{-14\pm4}{2}\Rightarrow x=-9,-5.

Q: What is the discriminant of x^{2}+14x+45 ?

\Delta=b^{2}-4ac=14^{2}-4(1)(45)=196-180=16.

Q: Find the vertex of y=x^{2}+14x+45 ?

Vertex at x=-\frac{b}{2a}=-7. y= (-7)^{2}+14(-7)+45=4. So (-7,4).

Answer

We factor the quadratic \(x^2 + 14x + 45\) by finding two numbers that add to \(14\) and multiply to \(45\). The numbers are \(9\) and \(5\).

\[
x^2 + 14x + 45 = (x+9)(x+5)
\]

Final result: \((x+9)(x+5)\).

Detailed Explanation

We are asked to simplify or factor the expression:

\[
x^2 + 14x + 45
\]

Step 1: Identify what factoring method to use.

This is a quadratic polynomial of the form \(ax^2 + bx + c\), where:

\[
a = 1,\quad b = 14,\quad c = 45
\]

Step 2: Use the AC (or factoring) method.

We want to find two numbers that:

Multiply to \(ac = 1 \cdot 45 = 45\)
Add to \(b = 14\)

Step 3: Find the pair of numbers.

We list factor pairs of \(45\):

\(1\) and \(45\) (sum \(46\))
\(3\) and \(15\) (sum \(18\))
\(5\) and \(9\) (sum \(14\))

The pair that adds to \(14\) is \(5\) and \(9\).

Step 4: Rewrite the quadratic as a product of two binomials.

Since the leading coefficient is \(1\), the factors are:

\[
x^2 + 14x + 45 = (x + 5)(x + 9)
\]

Final Answer:

\[
\boxed{(x + 5)(x + 9)}
\]

See full solution

Graph

Try AI Homework Help, get instant solutions!

Homework Helper

Algebra FAQ

Factor \(x^{2}+14x+45\) ?

\(x^{2}+14x+45=(x+5)(x+9)\).

What are the roots of \(x^{2}+14x+45=0\) ?

From \((x+5)(x+9)=0\), \(x=-5\) or \(x=-9\).

Complete the square for \(x^{2}+14x+45\) ?

\(x^{2}+14x+45=(x+7)^{2}-4\).

Solve using the quadratic formula for \(x^{2}+14x+45=0\) ?

\(a=1,b=14,c=45\). \(x=\frac{-14\pm\sqrt{196-180}}{2}=\frac{-14\pm4}{2}\Rightarrow x=-9,-5\).

What is the discriminant of \(x^{2}+14x+45\) ?

\(\Delta=b^{2}-4ac=14^{2}-4(1)(45)=196-180=16\).

Find the vertex of \(y=x^{2}+14x+45\) ?

Vertex at \(x=-\frac{b}{2a}=-7\). \(y= (-7)^{2}+14(-7)+45=4\). So \((-7,4)\).

Solve x²+14x+45 step by step.
Check answers with AI homework tools.

298,376+ active customers
Math, Geometry, Trigonometry, etc.

AI Math Solver Geometry AI Trig Solver