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Chapter 6: Problem 76

In Exercises \(67-78,\) factor completely. $$-5 x^{2}+30 x-45$$

Short Answer

Expert verified
The factored form of the expression is \(-5(x-3)^{2}\).

Step by step solution

01

Factor out a common factor

The given expression is \(-5 x^{2}+30 x-45\). Look at each of the coefficients (the numbers before each variable). They are -5, 30 and -45. The greatest common factor among these numbers is 5. So, factor out 5 from each term to get \(-5(x^{2}-6x+9)\).
02

Factor the quadratic

The expression \(x^{2}-6x+9\) is a quadratic expression, and it can be factored further. Looking at the numbers, this is a perfect square trinomial, because \((3)^{2}=9,\) and \(2*3=6.\) So, this can be factored as \((x-3)^{2}\)
03

Write the final answer

Combine the factored out common factor and the factored quadratic to get the final factored form of the expression. So, the fully factored expression is \(-5(x-3)^{2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
To solve polynomial expressions like \(-5 x^{2}+30 x-45\), we first look for the common factor. A common factor is a number or expression that divides each term in the polynomial without leaving a remainder. In this case, the coefficients are -5, 30, and -45.
  • Identify the greatest common factor among these numbers.
  • Here, the greatest common factor is 5.
When extracting a common factor, you divide each term in the polynomial by this factor and place it outside of the parentheses. This transforms \(-5x^2 + 30x - 45\) into \(-5(x^2 - 6x + 9)\).Finding and factoring out the common factor simplifies the expression, making it easier to tackle subsequently.
Quadratic Expression
Quadratic expressions have a characteristic form: \(ax^2 + bx + c\). In the expression \(x^2 - 6x + 9\), the quadratic expression has a coefficient \(a = 1\), \(b = -6\), and \(c = 9\). These types of polynomials are defined by a degree of 2, meaning the highest exponent of the variable is 2.
Quadratic expressions are common and have various methods of factorization including:
  • factoring by grouping,
  • using the quadratic formula,
  • completing the square.
For most cases like the one above, simple factoring methods can lead you to the solution faster.
Being familiar with different forms and structures allows us to spot special cases, like a perfect square trinomial, more easily.
Perfect Square Trinomial
A perfect square trinomial is a special form of quadratic expression \(a^2 \pm 2ab + b^2\), where the expression can be simplified into \((a \pm b)^2\). In our example, \(x^2 - 6x + 9\), it fits this form perfectly:
  • Here, \(a = x\) and \(b = 3\).
  • We recognize it because \((3)^2 = 9\) and \(2 \cdot 3 \cdot x = 6x\).
Thus, the expression \(x^2 - 6x + 9\) simplifies into \((x - 3)^2\). Recognizing this form is crucial because it allows for quick and easy factorization, reducing computation and simplifying the polynomial to its most concise form.

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Most popular questions from this chapter

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &3 x^{3}-12 x^{2}-15 x=3 x(x+5)(x-1) ;[-5,7,1] \text { by }\\\ &[-80,80,10] \end{aligned}$$

In Exercises \(119-122\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\begin{array}{ccccccc}\text { All } & \text { perfect } & \text { square } & \text { trinomials } & \text { are } & \text { squares } & \text { of }\end{array}\) binomials.

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(4 y^{2}+44 y+121=0\)

A ball is thrown straight up from a rooftop 300 feet high. The formula $$h=-16 t^{2}+20 t+300$$ describes the ball's height above the ground, \(h\), in feet, t seconds after it was thrown. The ball misses the rooftop on its way down and eventually strikes the ground. The graph of the formula is shown, with tick marks omitted along the horizontal axis. Use the formula to solve. When will the ball's height be 304 feet? Identify the solution as a point on the graph.

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. The height, in feet, of the camera after \(t\) seconds is modeled by the polynomial \(400-16 t^{2}\). The factored form of the polynomial is \(16(5+t)(5-t) .\) Describe something about your falling camera that is casier to sec from the factored form, \(16(5+t)(5-t),\) than from the form \(400-16 t^{2}\)
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