Question

Write the quadratic equation in standard form:
4, x, squared, plus, 8, equals, minus, 5, x
4x
2
+8=
−5x
Answer
Write the quadratic equation in standard form:
8, x, squared, minus, 8, x, minus, 15, equals, 5, x, squared
8x
2
−8x−15=
5x
2
Answer
Factor the expression completely.
20, x, to the power 4 , plus, 45, x, to the power 5
20x
4
+45x
5
Answer
Factor the expression completely.
36, x, to the power 4 , minus, 28, x, squared
36x
4
−28x
2
Answer
Solve the equation for all values of xx.
x, squared, minus, 49, equals, 0
x
2
−49=
0
Answer
Solve the equation for all values of xx.
x, squared, minus, 121, equals, 0
x
2
−121=
0
Answered by GPT-4o mini
To write the given equation \( 4x^2 + 8 = -5x \) in standard form, we need to rearrange the equation so that one side equals zero.

Starting with the original equation:

\[
4x^2 + 8 = -5x
\]

We can add \( 5x \) to both sides to move everything to one side of the equation:

\[
4x^2 + 5x + 8 = 0
\]

Now, the equation is in standard form, which is:

\[
4x^2 + 5x + 8 = 0
\]
Answered by GPT-4o mini
To write the equation \( 8x^2 - 8x - 15 = 5x^2 \) in standard form, we need to rearrange it so that all terms are on one side of the equation, leaving the other side equal to zero.

Starting with:

\[
8x^2 - 8x - 15 = 5x^2
\]

We can subtract \( 5x^2 \) from both sides:

\[
8x^2 - 5x^2 - 8x - 15 = 0
\]

Now combine like terms:

\[
(8x^2 - 5x^2) - 8x - 15 = 0
\]

This simplifies to:

\[
3x^2 - 8x - 15 = 0
\]

So the quadratic equation in standard form is:

\[
3x^2 - 8x - 15 = 0
\]
Answered by GPT-4o mini
To factor the expression \( 20x^4 + 45x^5 \) completely, we can start by identifying the greatest common factor (GCF) of the two terms.

1. **Identify the GCF**:
- The coefficients are 20 and 45. The GCF of 20 and 45 is 5.
- For the variable part, the lowest power of \( x \) in both terms is \( x^4 \).

So, the GCF of the entire expression is \( 5x^4 \).

2. **Factor out the GCF**:

Now, we can factor out \( 5x^4 \) from the original expression:

\[
20x^4 + 45x^5 = 5x^4(4 + 9x)
\]

3. **Check for further factorization**:

The remaining expression \( 4 + 9x \) cannot be factored further in terms of integers or simpler polynomials.

Thus, the completely factored form of the expression is:

\[
\boxed{5x^4(4 + 9x)}
\]
Answered by GPT-4o mini
To factor the expression \( 36x^4 - 28x^2 \) completely, we will follow these steps:

1. **Identify the Greatest Common Factor (GCF)**:
- The coefficients are 36 and 28. The GCF of 36 and 28 is 4.
- For the variable part, the lowest power of \( x \) in both terms is \( x^2 \).

So, the overall GCF of \( 36x^4 - 28x^2 \) is \( 4x^2 \).

2. **Factor out the GCF**:

Now, we can factor out \( 4x^2 \):

\[
36x^4 - 28x^2 = 4x^2(9x^2 - 7)
\]

3. **Check for further factorization**:

Next, we look at the expression inside the parentheses, \( 9x^2 - 7 \). This is a difference of squares format but is not factorable into integer coefficients.

Thus, the completely factored form of the expression is:

\[
\boxed{4x^2(9x^2 - 7)}
\]
Answered by GPT-4o mini
To solve the equation \( x^2 - 49 = 0 \), we can factor the left-hand side. Notice that this is a difference of squares:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

Setting the factored equation equal to zero gives us:

\[
(x - 7)(x + 7) = 0
\]

Now, we can set each factor to zero:

1. \( x - 7 = 0 \)
\[
x = 7
\]

2. \( x + 7 = 0 \)
\[
x = -7
\]

Thus, the solutions to the equation \( x^2 - 49 = 0 \) are:

\[
\boxed{7} \quad \text{and} \quad \boxed{-7}
\]
Answered by GPT-4o mini
To solve the equation \( x^2 - 121 = 0 \), we can recognize that this is also a difference of squares:

\[
x^2 - 121 = (x - 11)(x + 11)
\]

Now, we can set each factor equal to zero:

1. \( x - 11 = 0 \)
\[
x = 11
\]

2. \( x + 11 = 0 \)
\[
x = -11
\]

Thus, the solutions to the equation \( x^2 - 121 = 0 \) are:

\[
\boxed{11} \quad \text{and} \quad \boxed{-11}
\]