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Write the quadratic equation in standard form:

minus, 4, x, minus, 19, equals, x, squared
−4x−19=
x
2

11 answers

To write the given equation \(-4x - 19 = x^2\) in standard form, we need to rearrange it so that all terms are on one side of the equation and set equal to zero.

Starting from the equation:

\[ -4x - 19 = x^2 \]

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

\[ -4x - 19 - x^2 = 0 \]

Now, rearranging the equation gives us:

\[ -x^2 - 4x - 19 = 0 \]

To write it in the standard form of \(ax^2 + bx + c = 0\), we can multiply the entire equation by \(-1\) to make the leading coefficient positive:

\[ x^2 + 4x + 19 = 0 \]

So, the quadratic equation in standard form is:

\[ x^2 + 4x + 19 = 0 \]

What is the greatest common factor of 24 and 36?

To find the greatest common factor (GCF) of 24 and 36, we can use the method of prime factorization.

  1. Prime factorization of 24:

    • \(24 = 2 \times 12\)
    • \(12 = 2 \times 6\)
    • \(6 = 2 \times 3\)
    • Thus, the prime factorization of 24 is: \[ 24 = 2^3 \times 3^1 \]

  2. Prime factorization of 36:

    • \(36 = 2 \times 18\)
    • \(18 = 2 \times 9\)
    • \(9 = 3 \times 3\)
    • Thus, the prime factorization of 36 is: \[ 36 = 2^2 \times 3^2 \]

  3. Finding the GCF:

    • Now, identify the lowest power of each common prime factor:
      • For \(2\), the lowest power is \(2^2\).
      • For \(3\), the lowest power is \(3^1\).

  4. Calculating the GCF: \[ GCF = 2^2 \times 3^1 = 4 \times 3 = 12 \]

Thus, the greatest common factor of 24 and 36 is 12.

Factor the expression completely.
15, plus, 20, x
15+20x

To factor the expression \(15 + 20x\) completely, we first look for the greatest common factor (GCF) of the terms.

  1. Identify the GCF of \(15\) and \(20x\):

    • The GCF of \(15\) and \(20\) is \(5\).

  2. Factor out the GCF:

    • We can factor \(5\) out of both terms: \[ 15 + 20x = 5(3) + 5(4x) \]
    • This simplifies to: \[ 5(3 + 4x) \]

Thus, the completely factored form of the expression \(15 + 20x\) is

\[ \boxed{5(3 + 4x)} \]

Factor the expression completely.
5, x, minus, x, squared
5x−x
2

To factor the expression \(5x - x^2\) completely, follow these steps:

  1. Rearrange the expression: Rewrite it in standard form: \[ -x^2 + 5x \] (It is sometimes easier to factor if the \(x^2\) term is first.)

  2. Factor out the common factor: We see that both terms \(5x\) and \(-x^2\) have a common factor of \(x\): \[ -x^2 + 5x = -x(x - 5) \]

Now we have factored out \(-x\) from the expression, and what remains is \((x - 5)\).

Thus, the completely factored form of the expression \(5x - x^2\) is: \[ \boxed{-x(x - 5)} \]

Solve for all values of xx by factoring.
x, squared, equals, minus, 2, x
x
2
=
−2x

To solve the equation \(x^2 = -2x\) by factoring, we first rearrange the equation to set it to zero:

\[ x^2 + 2x = 0 \]

Next, we can factor out the common term \(x\):

\[ x(x + 2) = 0 \]

Now, we can apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Setting each factor to zero gives us:

  1. \(x = 0\)
  2. \(x + 2 = 0 \Rightarrow x = -2\)

Thus, the solutions to the equation are:

\[ \boxed{0 \text{ and } -2} \]

Solve the equation for all values of xx.
x, squared, minus, 9, equals, 0
x
2
−9=
0

To solve the equation

\[ x^2 - 9 = 0, \]

we can factor it. The expression \(x^2 - 9\) is a difference of squares, which can be factored as follows:

\[ x^2 - 9 = (x - 3)(x + 3). \]

Now we set the factored equation to zero:

\[ (x - 3)(x + 3) = 0. \]

Using the zero product property, we can set each factor to zero:

  1. \(x - 3 = 0 \Rightarrow x = 3\)
  2. \(x + 3 = 0 \Rightarrow x = -3\)

Thus, the solutions to the equation \(x^2 - 9 = 0\) are:

\[ \boxed{3 \text{ and } -3}. \]