Recognize Perfect Square Expressions
Read this topicExplanations 3
- What Are They? A perfect square quadratic equation is just a quadratic equation that can be factored into the product of two of the same binomial. For example, (2x+4)2(2x+4)^2(2…
By Andrew Wang, Lili Sun, Alex Federspiel
- So, what exactly is a "perfect square"? A perfect square is the product of two identical terms multiplied together. Here are some examples: 1⋅1=12⋅2=43⋅3=94⋅4=165⋅5=25\begin{ali…
By Hannah Bonville, Alex Federspiel, Victor Wang, Evan Fang
- We say an integer is a perfect square if it is equal to the product of two of the same integer. For example 444 is a perfect square because 2⋅2=42 \cdot 2 = 42⋅2=4 and 169169169…
By Evan Fang
Problems 38
- How many distinct roots does 16x2−72x+8116x^2-72x+8116x2−72x+81 have?
- Factor 4x2+12x+94x^2+12x+94x2+12x+9.
- What are the real roots to the equation 49x2−42x+49=4049x^2 - 42x + 49 = 4049x2−42x+49=40? Write your answer(s) as a common fraction with no spaces.
- Solve for xxx. ∣x2−6x+9∣>0|x^2-6x+9| > 0∣x2−6x+9∣>0
- Find the unique positive solution to the equation 14x2−7x+49=0\frac{1}{4}x^2-7x+49 = 041x2−7x+49=0.
- Find the root(s) of y(x)=25x2−100x+100y(x)=25x^2-100x+100y(x)=25x2−100x+100.
- Find the solutions to 144x2−40x+18=32x+9144x^2-40x+18=32x+9144x2−40x+18=32x+9.
- What is the leading coefficient of the expanded version of (9x+2)2(9x+2)^2(9x+2)2?
- What is the constant term of the expanded version of (−4x−7)2(-4x-7)^2(−4x−7)2?
- The standard form for an expanded perfect square quadratic is which of the following?
- A coin that comes up heads with probability p>0p > 0p>0 and tails with probability 1−p>01-p > 01−p>0 independently on each flip is flipped eight times. Suppose the probability o…
- How many roots can a quadratic of the form y(x)=(ax+b)2y(x) = (ax+b)^2y(x)=(ax+b)2 have?
- Solve for xxx in the equation ∣x∣2+2∣x∣+1=0|x|^2+2|x|+1=0∣x∣2+2∣x∣+1=0. Select all that apply.
- If (ax+12)2(ax+12)^2(ax+12)2 is a quadratic expression in factored form with some constant a,a,a, what is the constant term in the expanded expression?
- If (4x+b)2(4x+b)^2(4x+b)2 is a quadratic expression in factored form with some constant bbb, what is the quadratic term in the expanded expression?
- If the quadratic term of a quadratic expression is 4x24x^24x2, which of the following could be the factored form of the expression? (Select all that apply.)
- If the constant term of a quadratic expression is 646464, which of the following could be the factored form of the expression? (Select all that apply.)
- If the linear term of a quadratic expression is 32x32x32x, which of the following could be the factored form of the expression? (Select all that apply.)
- If the linear term of a quadratic expression is −64x-64x−64x, which of the following could be the factored form of the expression? (Select all that apply.)
- Determine which of the following quadratic expressions are perfect squares. (Select all that apply.)\\ Hint: these expressions will have the form a2x2+2abx+b2a^2x^2+2abx+b^2a2x2…
- Determine which of the following quadratic expressions are perfect squares. (Select all that apply.)\\ Hint: these expressions will have the form a2x2+2abx+b2a^2x^2+2abx+b^2a2x2…
- Determine which of the following quadratic expressions are perfect squares. (Select all that apply.)\\ Hint: these expressions will have the form a2x2+2abx+b2a^2x^2+2abx+b^2a2x2…
- Use intuition, previous information, and math tricks to help evaluate the following root, WITHOUT a calculator. 100040004\sqrt{100040004}100040004
- Select every equation below in which one side is a perfect square quadratic (in any form), while the other side is a constant.
- Select every equation below in which one side is a perfect square quadratic (in any form), while the other side is a constant.
- If (x+1)2=0(x+1)^2 = 0(x+1)2=0, then which value of xxx solves the equation?
- If 2x2−4x+2=02x^2-4x+2 = 02x2−4x+2=0, then which value of xxx solves the equation?
- If x2+4=4xx^2+4 = 4xx2+4=4x, then which value of xxx solves the equation?
- If x2+2x+1=3x^2+2x+1 = 3x2+2x+1=3, then the possible real values of xxx are...
- If 2x2+8x+8=122x^2+8x+8 = 122x2+8x+8=12, then the possible real values of xxx are...
- Which equation below shows a perfect square quadratic being set equal to zero? Assume: aaa, bbb, and ccc are constants.
- x2+2x+1=0x^2 + 2x + 1 = 0x2+2x+1=0 Is this a perfect square?
- 4x2+4x+1=04x^2+ 4x + 1 = 04x2+4x+1=0 Is the left hand expression, 4x2+4x+14x^2+4x+14x2+4x+1, a perfect square?
- x2+6x+6=0x^2+ 6x + 6 = 0x2+6x+6=0 Is x2+6x+6x^2+ 6x + 6x2+6x+6, the expression on the left, a perfect square?
- 9x2+x+9=09x^2+ x + 9 = 09x2+x+9=0 Is the expression on the left a perfect square?
- If a2x2+2abx+b2=0,a^2x^2 + 2abx + b^2 = 0,a2x2+2abx+b2=0, what would be the value(s) of xxx? Assume aaa is nonzero.
- The left hand side of 4x2+20x+25=04x^2 + 20x + 25 = 04x2+20x+25=0 is a perfect square expression. What value of xxx makes the equation true?
- Is 9x2−30x+259x^2 - 30x + 259x2−30x+25 a perfect square trinomial?