Introduction The aim of this investigation is to identify the nature of the roots of quadratics and cubic functions. Part One Case One For Case One‚ the discriminant of the quadratic will always be equal to zero. This will result in the parabola cutting the axis once‚ or twice in the same place‚ creating a distinct root or two of the same root. For PROOF 1‚ the equation y=a(x-b)2 is used. PROOF 1 y = 3 (x – 2)2 = y = 3 (x2 – 4x + 4) = y = 3x2 – 12x + 12 ^ = b2 – 4ac = (-12)2 – 4 x 3 x
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Name: ________________________ Date:__________________ Year & Section: _________________ Teacher: _______________ Reviewer: Quadratic Equations I. Multiple Choice: Choose the letter of the correct answer. Show your solution. 1. What are the values of x that satisfy the equation 3 – 27x2 = 0? A. x = [pic]3 B. x = [pic] C. x = [pic] D. x = [pic] 2. What are the solutions of the equation 6x2 + 9x – 15 = 0? A. 1‚ - 15 B. 1‚ [pic] C. – 1‚ - 5 D. 3‚ [pic]
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MCR3U0: Unit 2 – Equivalent Expressions and Quadratic Functions Radical Expressions 1) Express as a mixed radical in simplest form. a) c) b) e) d) f) 2) Simplify. a) b) d) e) c) f) 3) Simplify. a) b) c) d) e) f) 4) Simplify. a) d) b) e) f) c) For questions 5 to 9‚ calculate the exact values and express your answers in simplest radical form. 5) Calculate the length of the diagonal of a square with side length 4 cm. 6) A square has an area of 450 cm2. Calculate the side length. 7) Determine
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QUADRATIC EQUATIONS Quadratic equations Any equation of the form ax2 + bx + c=0‚ where a‚b‚c are real numbers‚ a 0 is a quadratic equation. For example‚ 2x2 -3x+1=0 is quadratic equation in variable x. SOLVING A QUADRATIC EQUATION 1.Factorisation A real number a is said to be a root of the quadratic equation ax2 + bx + c=0‚ if aa2+ba+c=0. If we can factorise ax2 + bx + c=0‚ a 0‚ into a product of linear factors‚ then the roots of the quadratic equation ax2 + bx + c=0 can be found
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Quadratic Equation: Quadratic equations have many applications in the arts and sciences‚ business‚ economics‚ medicine and engineering. Quadratic Equation is a second-order polynomial equation in a single variable x. A general quadratic equation is: ax2 + bx + c = 0‚ Where‚ x is an unknown variable a‚ b‚ and c are constants (Not equal to zero) Special Forms: * x² = n if n < 0‚ then x has no real value * x² = n if n > 0‚ then x = ± n * ax² + bx = 0 x = 0‚ x = -b/a
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This article is about quadratic equations and solutions. For more general information about quadratic functions‚ see Quadratic function. For more information about quadratic polynomials‚ see Quadratic polynomial. A quartic equation is a fourth-order polynomial equation of the form. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Monomial – is a polynomial with only one term. Binomial
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329 Quadratic Equations Chapter-15 Quadratic Equations Important Definitions and Related Concepts 1. Quadratic Equation If p(x) is a quadratic polynomial‚ then p(x) = 0 is called a quadratic equation. The general formula of a quadratic equation is ax 2 + bx + c = 0; where a‚ b‚ c are real numbers and a 0. For example‚ x2 – 6x + 4 = 0 is a quadratic equation. 2. Roots of a Quadratic Equation Let p(x) = 0 be a quadratic equation‚ then the values of x satisfying p(x) = 0 are called its roots or
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Pythagorean Quadratic Member MAT 222 Introduction to Algebra Instructor Yvette Gonzalez-Smith August 04‚ 2013 Pythagorean Quadratic The Pythagorean Theorem is an equation that allows a person to find the length of a side of a right triangle‚ as long as the length of the other two sides is known. The theorem basically relates the lengths of three sides of any right triangle. The theorem states that the square of the hypotenuse is the sum of the squares of the legs
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Pythagorean Quadratic MAT 221: Introduction to Algebra Pythagorean Quadratic The Pythagorean Theorem was termed after Pythagoras‚ who was a well-known Greek philosopher and mathematician‚ and the Pythagorean Theorem is one of the first theorems identified in ancient civilizations. “The Pythagorean theorem says that in any right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse” (Dugopolski‚ 2012‚ p. 366 para. 8). For this reason
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both sides of equation‚ which then leaves us with: x2 +16 = 8x + 36. The next step would then be to subtract 36 from both sides to get a result of. x2 -20= 8x. Finally we need to subtract 8x from both sides to get x2 – 8x – 20 =0. Now we have a quadratic equation to solve by factoring and using the zero factor. (x – ) (x + ) = 0 Since the coefficient of x2 is 1 we can start with a pair of parenthesis with an x in each. Since the 20 is negative we know there will be one + and one – in the binomials
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