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Quadratic equations are an essential part of algebra, and we must be familiar with them and the methods of solving quadratic equation questions.\u00a0<\/span><\/p>\n Before solving quadratic equation questions, let us have a quick recap of the quadratic equation concept and formulas.<\/span><\/p>\n <\/span><\/p>\n A quadratic equation is represented in the form of (ax\u00b2 + bx + c) where x is the variable, and a, b and c are real numbers or constants. It is a polynomial in which the highest power of the variable is 2.\u00a0<\/span><\/p>\n In the quadratic equation\u2019s standard form, the value of a can not be zero. If \u2018a\u2019 is zero, it will remove the \u2018<\/span>x\u00b2\u2019 term, and the equation will not remain quadratic.<\/p>\n Some examples of quadratic equations are:<\/span><\/b><\/span><\/p>\n 45x\u00b2 + 3x + 4, where a = 45, b = 3 and c = 4 (when we compare it with ax\u00b2 + bx + c).<\/span><\/b><\/span><\/p>\n Similarly, in the quadratic equation (- 4x\u00b2 + 54x – 40), we have a = – 4, b = 54 and c = – 40.<\/span><\/p>\n <\/span><\/p>\n The roots of the quadratic equation are the values of x for which ax\u00b2 + bx + c = 0. Since the degree of a quadratic equation is 2, we obtain two roots.<\/span><\/p>\n <\/span><\/p>\n The quadratic equation formula is also known as the Sridharacharya Formula. It is a method to find the roots of the equation. This formula is given below:<\/span><\/p>\nx=\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}<\/span>\n (<\/span>b\u00b2 – 4ac) is called the discriminant of the equation and is represented by D.<\/span><\/p>\n Therefore D = (b\u00b2 – 4ac)<\/span><\/p>\n We can define the nature of roots based on the discriminant value. There are 3 possible conditions given below:<\/span><\/p>\n <\/span><\/p>\n Now let us see some quadratic equation problems and their solutions.<\/span><\/p>\n <\/span><\/p>\n 1. Check if x(x + 2) + 6 = (x + 2) (x \u2013 2) is in the form of quadratic equation.<\/b><\/p>\n Solution:<\/b> Given,<\/span><\/p>\n x(x + 2) + 6 = (x + 2) (x \u2013 2)<\/span><\/p>\n \u21d2 x\u00b2 + 2x + 6 = x\u00b2 – 2\u00b2\u00a0 \u00a0 [By algebraic identities]<\/span><\/p>\n On Simplifying the equation<\/span><\/p>\n 2x + 6 = – 4<\/span><\/p>\n \u21d2 2x + 10 = 0<\/span><\/p>\n Since this equation is not in the form of ax\u00b2 + bx + c, it is not a quadratic equation.<\/span><\/p>\n <\/p>\n 2. Find the roots of the equation <\/b>2x\u00b2 – 8x + 8<\/strong> = 0 using factorisation.<\/b><\/p>\n Solution:<\/b>\u00a0 Given,<\/span><\/p>\n 2x\u00b2 – 8x + 8 = 0<\/p>\n \u21d2 <\/span>2x\u00b2 – 4x – 4x + 8 = 0<\/p>\n \u21d2 2x(x – 2) – 4(x – 2) = 0<\/span><\/p>\n \u21d2 (2x – 4) (x – 2) = 0<\/span><\/p>\n So, (2x – 4) = 0, and (x – 2) = 0<\/span><\/p>\n If 2x – 4 = 0; x = 2,<\/span><\/p>\n and if (x – 2) = 0, we get x = 2<\/span><\/p>\n Therefore, there are two equal roots of the given equation, and the root is 2.<\/span><\/p>\n <\/span><\/p>\n 3. Solve the quadratic equation x\u00b2 + 4x – 5<\/strong> = 0.<\/b><\/p>\n Solution:<\/b><\/p>\n x\u00b2 + 4x – 5 = 0<\/p>\n \u21d2 <\/span>x\u00b2 – x + 5x – 5 = 0<\/p>\n \u21d2 x(x – 1) + 5(x – 1) = 0<\/span><\/p>\n \u21d2 (x – 1)(x + 5) = 0<\/span><\/p>\n So, (x – 1) = 0 and (x + 5) = 0.<\/span><\/p>\n If x – 1 = 0; x = 1.<\/span><\/p>\n And if x + 5 = 0; x = -5<\/span><\/p>\n Therefore, the roots of the given quadratic equation are -5 and 1.<\/span><\/p>\n <\/p>\n 4. Solve the quadratic equation 2x\u00b2 + x – 528 = 0<\/b>, using quadratic formula.<\/b><\/p>\n Solution<\/b>: Comparing the equation with standard equation ax\u00b2 + bx + c = 0.<\/span><\/p>\n We find, a = 2, b = 1, and c = -528.<\/span><\/p>\n By using the quadratic formula:<\/span><\/p>\n x=\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}<\/span><\/span><\/p>\n x=\\frac{-1 \\pm \\sqrt{1^{2}-(4\\times{(2)}\\times{(-528))}}}{2 \\times 2}<\/span><\/span><\/p>\n \u21d2 x=\\frac{-1 \\pm \\sqrt{1-(4\\times(2)\\times (-528))}}{4}<\/span><\/span><\/p>\n \u21d2 x=\\frac{-1 \\pm \\sqrt{1+4224}}{4}<\/span><\/span><\/p>\n \u21d2 x=\\frac{-1 \\pm \\sqrt{4225}}{4}<\/span><\/span><\/p>\n \u21d2 x=\\frac{-1 \\pm 65}{4}<\/span><\/span><\/p>\n So, x=\\frac{-1 + 65}{4}\\text{ and }x=\\frac{-1-65}{4}<\/span><\/span><\/p>\n \u21d2 x=\\frac{64}{4}\\text{ and }x=\\frac{-66}{4}<\/span><\/span><\/p>\n \u21d2 x=16\\text{ and }x=\\frac{-33}{2}<\/span><\/span><\/p>\n Therefore, the roots of the equation are 16 and -33\/2.<\/span><\/p>\n <\/p>\n 5. What is the discriminant of the equation 3x\u00b2 – 2x +1\/3\u00a0 = 0.<\/b><\/p>\n Solution:<\/b> Comparing the equation with standard equation ax\u00b2 + bx + c = 0,\u00a0<\/span><\/p>\n a = 3, b = – 2, and c = \u2153.<\/span><\/p>\n Formula for discriminant, D = b\u00b2 – 4ac<\/span><\/p>\n D = (-2)\u00b2 – 4<\/span>\u00d7(3)\u00d7(\u2153)\u00a0<\/span><\/p>\n \u21d2 D = 4 – 4<\/span><\/p>\n \u21d2 D = 0<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n Solving this quadratic equation-\u00a0<\/span><\/p>\nx^{2}-36 x-9 x+324=0<\/span>\n \u21d2 x(x-36)-9(x-36)=0<\/span><\/p>\n \u21d2 (x-9)(x-36)=0<\/span><\/p>\n So, x<\/span> = 9 and 36.<\/span><\/p>\n <\/span><\/p>\n The given polynomial or quadratic equation is<\/span><\/p>\n x\u00b2 + 5x + 7 = 21<\/span><\/p>\n Using factorization method,\u00a0<\/span><\/p>\n x\u00b2 + 5x + 7 – 21 = 0<\/span><\/p>\n \u21d2 x\u00b2 + 5x – 14 = 0<\/span><\/p>\n \u21d2 x\u00b2 – 2x + 7x – 14 = 0<\/span><\/p>\n \u21d2 x(x – 2) + 7(x – 2) = 0<\/span><\/p>\n \u21d2 (x – 2)(x + 7) = 0<\/span><\/p>\n So, (x – 2) = 0, and (x + 7) = 0\u00a0<\/span><\/p>\n The roots (solution) of the quadratic equation, x = 2, -7.<\/span><\/p>\n <\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Row for space” _builder_version=”4.10.6″ _module_preset=”default” locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.11″ _module_preset=”default” global_colors_info=”{}”][et_pb_divider show_divider=”off” _builder_version=”4.10.4″ _module_preset=”default” global_colors_info=”{}”][\/et_pb_divider][\/et_pb_column][\/et_pb_row][\/et_pb_section][et_pb_section fb_built=”1″ admin_label=”banner and faq Section” module_class=”mainsec2″ _builder_version=”4.10.4″ _module_preset=”default” custom_padding=”40px||0px||false|false” global_colors_info=”{}”][et_pb_row use_custom_gutter=”on” gutter_width=”1″ make_equal=”on” disabled_on=”on|on|off” admin_label=”banner Row” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fff7d6″ width=”100%” max_width=”1310px” height=”134px” custom_margin=”||50px||false|false” custom_padding=”12px||12px||true|false” border_radii=”on|11px|11px|11px|11px” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_image src=”https:\/\/stgwebsite.mindspark.in\/wordpress\/wp-content\/uploads\/2021\/08\/calloutImage.png” title_text=”calloutImage” show_bottom_space=”off” admin_label=”Image” module_class=”img1″ _builder_version=”4.10.8″ _module_preset=”default” width=”25px” height=”60px” custom_padding=”2px||2px||true|false” global_colors_info=”{}”][\/et_pb_image][et_pb_text module_class=”ftstyle” _builder_version=”4.9.10″ _module_preset=”default” text_orientation=”center” global_colors_info=”{}”]<\/p>\nDefinition of Quadratic Equations<\/b><\/span><\/h2>\n
Roots of a Quadratic Equation<\/b><\/h2>\n
Quadratic Equation Formula<\/b><\/b><\/h2>\n
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Quadratic Equations Questions and Solutions<\/b><\/h2>\n
Examples<\/b><\/h2>\n
1. Find the roots of the quadratic equation x^{2}-45 x+324=0<\/span><\/span><\/h3>\nx^{2}-45 x+324=0<\/span>\n
2. Solve the quadratic equation x\u00b2 + 5x + 7 = 21.<\/span><\/h3>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nRelated Concepts<\/h1>\n