Questions and Answers
How many real solutions does the equation $9x²-3x+1=0$ have?
What are the zeros of the equation $y=x²+6x-27$?
What is the vertex of the equation $y=(x-1)²-5$?
What is the vertex of the equation $y=x²+4x+3$?
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What is the distance between the points (1,3) and (4,-1)?
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If a screen is 18 inches wide and has a screen size of 20 inches, how tall is it to the nearest whole inch?
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Factor the polynomial $8a²b²+10ab³c$.
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Factor the polynomial $2x³+2x²+x+1$.
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Factor the polynomial $2x²+6x-8$.
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Factor the polynomial $9x²+12x+4$.
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Factor the polynomial $6x²-7xy+2y²$.
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Graph the equation $y=x²+6x+5$. Show all work.
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Simplify the expression $√12x⁵$.
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What is the cube root of -8?
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What is the result of the expression $(√a+2)(√a+3)$?
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What is the simplified form of $4y / √2y$?
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What is the result of the expression $5 / (√7-1)$?
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Solve the equation $5x²-3=17$.
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Solve the quadratic equation $2x²+5x-3=0$.
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Solve the equation $x²-x=6$.
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Solve the equation $x²-2x-2=0$.
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Solve the equation $√x²-4x+13=x+7$.
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Study Notes
Multiple Choice Overview
- For the equation (9x²-3x+1=0), the discriminant (b²-4ac) results in (-27), indicating 0 real solutions.
- The zeros of the quadratic (y=x²+6x-27) are found to be (3,0) and (-9,0) by factoring as ((x-3)(x+9)=0).
Vertex Identification
- The vertex for (y=(x-1)²-5) is determined to be (1,-5) using the vertex form (y=(x-h)²+k).
- For the equation (y=x²+4x+3), the vertex is calculated as (-2,-1) through the formula (x=-b/2a).
Distance Calculation
- The distance between points (1,3) and (4,-1) is found using the distance formula, resulting in 5.
Geometry Problem
- Given an 18-inch wide screen with a diagonal of 20 inches, the height is approximately 9 inches after applying the Pythagorean theorem.
Polynomial Factoring
- The expression (8a²b²+10ab³c) factors to (2ab²(4a+5bc)).
- The polynomial (2x³+2x²+x+1) can be factored to ((x+1)(2x²+1)).
Simplification
- The expression (\sqrt{12x⁵}) simplifies to (2x²\sqrt{3x}).
- The cube root of (-8) evaluates to 2.
Completing Operations
- The expression ((\sqrt{a}+2)(\sqrt{a}+3)) expands to (a+5\sqrt{a}+6).
- The simplification of (\frac{4y}{\sqrt{2y}}) leads to (2\sqrt{2y}).
Rationalizing Denominators
- The expression (\frac{5}{\sqrt{7}-1}) simplifies to (\frac{5\sqrt{7}+5}{6}) through rationalization.
Solving Equations
- For the equation (5x²-3=17), the solutions are (x=±2) after rearranging and taking the square root.
- The quadratic (x²+5x-3=0) yields roots of (x=\frac{1}{2}) or (x=-3) using the quadratic formula.
Additional Equations
- The equation (x²-x=6) is factored to find solutions (x=-2) or (x=3).
- For (x²-2x-2=0), the solutions are (x=1±\sqrt{3}), derived from completing the square.
- The equation (\sqrt{x²-4x+13}=x+7) results in (x=-2), confirmed by verification of the solution.
Studying That Suits You
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Description
Prepare for the Abeka Grade 9 Algebra 1 Test 9 with these flashcards. This quiz covers key concepts including determining the number of real solutions to quadratic equations and identifying zeros of functions. Test your skills and deepen your understanding of essential algebra topics.
